Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2013 Article ID 324940 11 pageshttpdxdoiorg1011552013324940

Research ArticleConfidence Intervals for the Coefficient of Variation ina Normal Distribution with a Known Population Mean

Wararit Panichkitkosolkul

Department of Mathematics and Statistics Faculty of Science and TechnologyThammasat University PhathumThani 12120Thailand

Correspondence should be addressed to Wararit Panichkitkosolkul wararitmathstatscituacth

Received 23 July 2013 Accepted 25 September 2013

Academic Editor Shein-chung Chow

Copyright copy 2013 Wararit Panichkitkosolkul This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents three confidence intervals for the coefficient of variation in a normal distribution with a known populationmean One of the proposed confidence intervals is based on the normal approximation The other proposed confidence intervalsare the shortest-length confidence interval and the equal-tailed confidence interval AMonte Carlo simulation studywas conductedto compare the performance of the proposed confidence intervals with the existing confidence intervals Simulation results haveshown that all three proposed confidence intervals perform well in terms of coverage probability and expected length

1 Introduction

Thecoefficient of variation of a distribution is a dimensionlessnumber that quantifies the degree of variability relative tothe mean [1] It is a statistical measure for comparing thedispersion of several variables obtained by different unitsThepopulation coefficient of variation is defined as a ratio of thepopulation standard deviation (120590) to the populationmean (120583)given by 120581 = 120590120583 The typical sample estimate of 120581 is given as

120581 =119878

119883 (1)

where 119878 is the sample standard deviation the square root ofthe unbiased estimator of population variance and 119883 is thesample mean

The coefficient of variation has been widely used in manyareas such as science medicine engineering economics andothers For example the coefficient of variation has also beenemployed by Ahn [2] to analyze the uncertainty of faulttrees Gong and Li [3] assessed the strength of ceramicsby using the coefficient of variation Faber and Korn [4]applied the coefficient of variation as a way of including ameasure of variation in the mean synaptic response of thecentral nervous system The coefficient of variation has alsobeen used to assess the homogeneity of bone test samples

to help determine the effect of external treatments on theproperties of bones [5] Billings et al [6] used the coefficientof variation to study the impact of socioeconomic statuson hospital use in New York City In finance and actuarialscience the coefficient of variation can be used as a measureof relative risk and a test of the equality of the coefficientsof variation for two stocks [7] Furthermore Pyne et al [8]studied the variability of the competitive performance ofOlympic swimmers by using the coefficient of variation

Although the point estimator of the population coefficientof variation shown in (1) can be a useful statisticalmeasure itsconfidence interval is more useful than the point estimator Aconfidence interval provides much more information aboutthe population characteristic of interest than does a pointestimate (eg Smithson [9]Thompson [10] and Steiger [11])There are several approaches available for constructing theconfidence interval for 120581 McKay [12] proposed a confidenceinterval for 120581 based on the chi-square distribution thisconfidence interval works well when 120581 lt 033 [13ndash17]Later Vangel [18] proposed a new confidence interval for 120581which is called a modified McKayrsquos confidence interval Hisconfidence interval is based on an analysis of the distributionof a class of approximate pivotal quantities for the nor-mal coefficient of variation In addition modified McKayrsquosconfidence interval is closely related to McKayrsquos confidence

2 Journal of Probability and Statistics

interval but it is usually more accurate and nearly exactunder normality Panichkitkosolkul [19] modified McKayrsquosconfidence interval by replacing the sample coefficient ofvariation with the maximum likelihood estimator for anormal distribution Sharma and Krishna [20] introducedthe asymptotic distribution and confidence interval of thereciprocal of the coefficient of variation which does notrequire any assumptions about the population distribution tobe made Miller [21] discussed the approximate distributionof 120581 and proposed the approximate confidence interval for 120581in the case of a normal distributionTheperformance ofmanyconfidence intervals for 120581 obtained by McKayrsquos Millerrsquos andSharma-Krishnarsquos methods was compared under the samesimulation conditions by Ng [22]

Mahmoudvand and Hassani [23] proposed an approxi-mately unbiased estimator for 120581 in a normal distribution andalso used this estimator for constructing two approximateconfidence intervals for the coefficient of variation Theconfidence intervals for 120581 in normal and lognormal wereproposed by Koopmans et al [24] and Verrill [25] Buntaoand Niwitpong [26] also introduced an interval estimatingthe difference of the coefficient of variation for lognormaland delta-lognormal distributions Curto and Pinto [27] con-structed the confidence interval for 120581when random variablesare not independently and identically distributed Recentwork of Gulhar et al [28] has compared several confidenceintervals for estimating the population coefficient of variationbased on parametric nonparametric andmodifiedmethods

However the population mean may be known in severalphenomena The confidence intervals of the aforementionedauthors have not been used for estimating the populationcoefficient of variation for the normal distribution with aknown population mean Therefore our main aim in thispaper is to propose three confidence intervals for 120581 in anormal distribution with a known population mean

The organization of this paper is as follows In Section 2the theoretical background of the proposed confidence inter-vals is discussed The investigations of the performance ofthe proposed confidence interval through a Monte Carlosimulation study are presented in Section 3 A comparisonof the confidence intervals is also illustrated by using anempirical application in Section 4 Conclusions are providedin the final section

2 Theoretical Results

In this section the mean and variance of the estimator ofthe coefficient of variation in a normal distribution witha known population mean are considered In addition wewill introduce an unbiased estimator for the coefficient ofvariation obtain its variance and finally construct threeconfidence intervals normal approximation shortest-lengthand equal-tailed confidence intervals

If the population mean is known to be 1205830 then the

population coefficient of variation is given by 1205810= 120590120583

0 The

sample estimate of 1205810is

1205810=1198780

1205830

(2)

where 11987820= 119899minus1sum

119899

119894=1(119883119894minus 1205830)2 To find the expectation of (2)

we have to prove the following lemma

Lemma 1 Let 1198831 1198832 119883

119899be a random sample from

normal distribution with known mean 1205830and variance 1205902 and

let 11987820= 119899minus1sum

119899

119894=1(119883119894minus 1205830)2 Then

119864 (1198780) = 119888119899+1120590

V119886119903 (1198780) = (1 minus 119888

2

119899+1) 1205902

(3)

where 119888119899+1

= radic2119899(Γ((119899 + 1)2)Γ(1198992))

Proof of Lemma 1 By definition

1198782

0=1

119899

119899

sum119894=1

(119883119894minus 1205830)2=1205902

119899

119899

sum119894=1

1198852

119894 (4)

where 119885119894= (119883119894minus 1205830)120590 sim 119873(0 1)

Thus

11989911987820

1205902sim 1205942

119899 (5)

Let 1198782 = (119899 minus 1)minus1sum119899

119894=1(119883119894minus 119883)2 and 1198782

lowast=

119899minus1sum119899+1

119894=1(119883119894minus 119883)2 From Theorem B of Rice [29 page 197]

the distribution of (119899 minus 1)11987821205902 is central chi-square distribu-tion with 119899minus 1 degrees of freedom Similarly the distributionof 1198991198782lowast1205902 is central chi-square distribution with 119899 degrees of

freedom that is

1198991198782

lowast

1205902sim 1205942

119899 (6)

One can see that [30 page 181]

119864 (119878) = 119888119899120590 (7)

where 119888119899= radic2(119899 minus 1)(Γ(1198992)Γ((119899 minus 1)2))

Similarly

119864 (119878lowast) = 119888119899+1120590 (8)

where 119888119899+1

= radic2119899(Γ((119899 + 1)2)Γ(1198992))Equations (5) and (6) are equivalent Thus we obtain

119864(1198780) = 119864(119878

lowast) = 119888

119899+1120590 Next we will find the variance of

1198780

var (1198780) = 119864 (119878

2

0) minus [119864 (119878

0)]2= 1205902minus 1198882

119899+11205902= (1 minus 119888

2

119899+1) 1205902

(9)

By using Lemma 1 we can show that the mean andvariance of 120581

0are

119864 (1205810) =

119888119899+1

120590

1205830

= 119888119899+11205810 (10)

var (1205810) = (

1 minus 1198882119899+1

12058320

)1205902= (1 minus 119888

2

119899+1) 1205812

0 (11)

Journal of Probability and Statistics 3

Cov

erag

e pro

babi

litie

s

086

090

094

01 02 03 04 05 061205810

n = 5

MillerMcKayVangel

ApproxShortestEqual-tailed

(a)C

over

age p

roba

bilit

ies

01 02 03 04 05 061205810

096

092

088

n = 15

MillerMcKayVangel

ApproxShortestEqual-tailed

(b)

Cov

erag

e pro

babi

litie

s

01 02 03 04 05 061205810

098

096

094

092

090

n = 25

MillerMcKayVangel

ApproxShortestEqual-tailed

(c)

Cov

erag

e pro

babi

litie

s

01 02 03 04 05 061205810

098

096

094

092

090

n = 50

MillerMcKayVangel

ApproxShortestEqual-tailed

(d)

Figure 1 The estimated coverage probabilities of 90 confidence intervals for the coefficient of variation in a normal distribution with aknown population mean

Note that 119888119899+1

rarr 1 as 119899 rarr infin Therefore it follows that

lim119899rarrinfin

119864 (1205810) = 120581

0 (12)

It means that 1205810is asymptotically unbiased and asymptoti-

cally consistent for 1205810 From (10) the unbiased estimator of

1205810is

1205810=

1205810

119888119899+1

(13)

Using Lemma 1 the mean and variance of 1205810are given by

119864 (1205810) = 119864(

1205810

119888119899+1

) = 1205810 (14)

var (1205810) = var(

1205810

119888119899+1

) =1

1198882119899+1

var(1198780

1205830

)

=1

1198882119899+112058320

(1 minus 1198882

119899+1) 1205902= (

1 minus 1198882119899+1

1198882119899+1

)1205812

0

(15)

4 Journal of Probability and Statistics

Expe

cted

leng

ths

01 02 03 04 05 061205810

15

10

05

00

n = 5

MillerMcKayVangel

ApproxShortestEqual-tailed

(a)Ex

pect

ed le

ngth

s

01 02 03 04 05 061205810

00

02

04

06

08

n = 15

MillerMcKayVangel

ApproxShortestEqual-tailed

(b)

Expe

cted

leng

ths

01 02 03 04 05 061205810

05

04

03

02

01

00

n = 25

MillerMcKayVangel

ApproxShortestEqual-tailed

(c)

Expe

cted

leng

ths

01 02 03 04 05 061205810

030

020

010

000

n = 50

MillerMcKayVangel

ApproxShortestEqual-tailed

(d)

Figure 2The expected lengths of 90 confidence intervals for the coefficient of variation in a normal distribution with a known populationmean

Thus

lim119899rarrinfin

var (1205810) = 0 (16)

Hence 1205810is also asymptotically consistent for 120581

0 Next we

examine the accuracy of 1205810from another point view Let us

first consider the following theorem

Theorem 2 Let 1198831 1198832 119883

119899be a random sample from

a probability density function 119891(119909) which has unknownparameter 120579 If 120579 is an unbiased estimator of 120579 it can be shown

under very general conditions that the variance of 120579must satisfythe inequality

var (120579) ge 1

119899119864 (minus12059721205971205792 ln119891 (119909))=

1

119899119868 (120579) (17)

where 119868(120579) is the Fisher information This is known as theCramer-Rao inequality If var(120579) = 1(119899119868(120579)) the estimator120579 is said to be efficient

Proof of Theorem 2 See [31 pages 377ndash379]

Journal of Probability and Statistics 5

Weight

Freq

uenc

y

2500 3500 4500 5500

20

15

10

5

0

(a)

Weight

2000 3000 4000 5000 6000

Den

sity

6e minus 04

4e minus 04

2e minus 04

0e + 00

(b)

6000

5000

4000

3000

(c)

Theoretical quantiles

Sam

ple q

uant

iles

6000

5000

4000

3000

minus2 minus1 0 1 2

(d)

Figure 3 (a) Histogram (b) density plot (c) Box-and-Whisker plot and (d) normal quantile-quantile plot of the weights of 61 one-monthold infants

By setting 120579 = 1205810= 120590120583

0in Theorem 2 it is easy to show

that

var (0) ge

12058120

2119899 (18)

where 0is any unbiased estimator of 120581

0 This means that the

variance for the efficient estimator of 1205810is 120581202119899

From (15) we will show that (1minus1198882119899+1)1198882119899+1

rarr 1(2119899minus1)The asymptotic expansion of the gamma function ratio is [32]

Γ (119895 + (12))

Γ (119895)= radic119895(1 minus

1

8119895+

1

1281198952+ sdot sdot sdot ) (19)

Now if 119895 = 1198992 in (19) we have

119888119899+1

= radic2

119899

Γ ((119899 + 1) 2)

Γ (1198992)

= radic2

119899[radic

119899

2(1 minus

1

4119899+

1

321198992+ sdot sdot sdot )]

= 1 minus1

4119899+ o( 1

11989932)

(20)

Thus we obtain

1198882

119899+1= 1 minus

1

2119899+ o( 1

1198992)

1 minus 1198882119899+1

1198882119899+1

997888rarr1

2119899 minus 1

(21)

6 Journal of Probability and Statistics

Table 1 The values of 119886 and 119887 for the shortest-length confidence interval for 1205810

dfConfidence levels

090 095 099119886 119887 119886 119887 119886 119887

2 02065 125208 01015 151194 00200 2082643 05654 131532 03449 155897 01140 2098564 10200 141800 06918 165735 02937 2183715 15352 153498 11092 177432 05461 2298676 20930 165807 15776 189954 08567 2426187 26828 178391 20851 202863 12143 2560178 32981 191099 26235 215953 16107 2697499 39343 203848 31874 229118 20394 28364310 45883 216598 37729 242303 24958 29760211 52573 229325 43768 255476 29760 31158012 59397 242016 49967 268618 34771 32554313 66337 254666 56308 281717 39968 33947414 73382 267269 62776 294769 45329 35335815 80521 279825 69357 307770 50840 36719216 87745 292334 76042 320720 56487 38096817 95047 304796 82820 333619 62256 39468818 102421 317212 89685 346467 68139 40834719 109861 329585 96629 359266 74126 42195220 117362 341915 103647 372016 80209 43549821 124919 354205 110733 384720 86383 44898922 132530 366455 117882 397379 92640 46242623 140191 378668 125092 409995 98976 47581024 147899 390844 132357 422570 105385 48914425 155650 402986 139675 435105 111864 50242826 163443 415095 147043 447601 118408 51566527 171275 427171 154458 460060 125014 52885628 179144 439217 161917 472483 131678 54200229 187049 451234 169419 484872 138397 55510730 194987 463222 176961 497229 145170 56816940 275919 581755 254233 619217 215331 69680850 359012 698342 334085 738920 288879 82253460 443661 813479 415794 856914 364863 94606370 529501 927487 498923 973573 442711 106786780 616290 1040584 583183 1089153 522044 118827290 703860 1152925 668374 1203839 602597 1307514100 792086 1264628 754347 1317767 684177 1425771150 1240372 1816128 1192737 1879079 1103262 2006194200 1696646 2359748 1640642 2431025 1534834 2574375250 2158057 2898273 2094667 2976910 1974440 3134620300 2623132 3433155 2553057 3518461 2419776 3689185

Therefore var(1205810) rarr 1205812

0(2119899 minus 1) This means that 120581

0is

asymptotically efficient (see (18)) In the following sectionthree confidence intervals for 120581

0are proposed

21 Normal Approximation Confidence Interval Using thenormal approximate we have

119911 =1205810minus 1205810

var (1205810)=

1205810119888119899+1

minus 1205810

radic(1 minus 1198882119899+1) 120581201198882119899+1

=1205810minus 119888119899+11205810

1205810radic1 minus 1198882

119899+1

997888rarr 119873(0 1)

(22)

Journal of Probability and Statistics 7

Table 2 The estimated coverage probabilities and expected lengths of 90 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08499 09066 08858 09016 09023 08979 00555 00607 00582 00741 00587 00675010 08518 09069 08866 09024 09008 08988 01120 01237 01177 01482 01174 01349020 08524 09130 08960 09036 08990 09006 02315 02689 02457 02963 02347 02696033 08572 09258 09136 09038 08999 09001 04099 05872 04528 04895 03878 04453050 08664 09430 09321 09036 08994 09001 06959 12123 09360 07409 05869 06741067 08773 09578 09428 09031 09000 08992 10603 15764 16394 09947 07880 09050

10

005 08747 09031 08870 09020 08996 09006 00379 00396 00382 00431 00388 00416010 08792 09052 08899 09024 09001 09014 00765 00804 00773 00864 00778 00833020 08802 09135 09002 09013 08993 09001 01576 01686 01603 01726 01553 01664033 08899 09304 09202 09017 09021 09015 02778 03140 02893 02853 02566 02750050 08999 09527 09451 09007 09004 08995 04709 06575 05323 04329 03895 04174067 09129 09694 09600 09018 08999 08992 07128 14205 10257 05801 05218 05593

15

005 08846 09010 08870 09000 08989 08988 00307 00316 00306 00333 00311 00326010 08866 09065 08925 09011 09013 09001 00618 00638 00617 00666 00622 00652020 08913 09127 09007 09006 08993 08987 01271 01328 01275 01330 01242 01301033 09046 09308 09218 09012 09022 09013 02239 02418 02286 02200 02054 02151050 09150 09544 09477 09000 09004 08991 03787 04522 04087 03338 03116 03264067 09280 09725 09661 09010 09002 08999 05713 08900 07010 04466 04170 04367

25

005 08933 09056 08932 09029 09035 09021 00236 00240 00233 00247 00238 00244010 08948 09054 08939 09010 09014 09004 00475 00485 00471 00495 00475 00489020 09022 09146 09042 09028 09008 09021 00977 01003 00971 00988 00949 00976033 09144 09318 09238 09005 09016 09004 01716 01796 01727 01630 01566 01610050 09285 09548 09491 08976 08977 08978 02893 03207 03027 02471 02374 02440067 09430 09768 09722 09018 08999 09008 04363 05418 04930 03310 03179 03268

50

005 08941 08992 08905 08993 08977 08989 00166 00167 00164 00170 00166 00168010 08996 09043 08949 09004 09007 08997 00334 00337 00330 00339 00333 00337020 09061 09118 09041 08994 08989 08996 00688 00697 00680 00678 00665 00674033 09220 09314 09253 08997 08996 08994 01206 01236 01204 01118 01096 01112050 09436 09583 09539 09009 09022 09010 02031 02153 02084 01695 01662 01685067 09588 09801 09770 09010 09009 09008 03062 03460 03309 02271 02226 02257

100

005 08998 09026 08961 09019 08902 09017 00117 00117 00115 00118 00113 00118010 09000 09031 08959 08995 08878 08992 00236 00237 00233 00236 00227 00236020 09110 09131 09072 09011 08901 09008 00485 00488 00480 00472 00453 00471033 09277 09329 09282 09015 08900 09012 00850 00863 00847 00779 00748 00777050 09485 09589 09561 09001 08885 08998 01429 01486 01455 01180 01133 01177067 09678 09810 09790 09020 08910 09021 02157 02347 02289 01582 01519 01578

Therefore the 100(1 minus 120572) confidence interval for 1205810based

on (22) is

1205810

119888119899+1

+ 1199111minus1205722

radic1 minus 1198882119899+1

le 1205810le

1205810

119888119899+1

minus 1199111minus1205722

radic1 minus 1198882119899+1

(23)

where 1199111minus1205722

is the 100(1 minus 1205722) percentile of the standardnormal distribution

22 Shortest-Length Confidence Interval A pivotal quantityfor 1205902 is

119876 =11989911987820

1205902sim 1205942

119899 (24)

Converting the statement

119875(119886 le11989911987820

1205902le 119887) = 1 minus 120572 (25)

we can write

119875(1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886) = 1 minus 120572 (26)

8 Journal of Probability and Statistics

Table 3 The estimated coverage probabilities and expected lengths of 95 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08829 09533 09440 09538 09504 09511 00661 00785 00762 01058 00758 00870010 08827 09537 09457 09549 09501 09506 01333 01608 01544 02113 01513 01737020 08847 09578 09508 09548 09500 09507 02756 03630 03282 04226 03026 03475033 08904 09647 09599 09542 09501 09501 04880 08954 06498 06986 05001 05743050 08934 09711 09656 09537 09487 09491 08276 13796 14333 10561 07561 08683067 09042 09795 09721 09548 09495 09502 12550 15758 19791 14140 10124 11625

10

005 09115 09522 09440 09511 09502 09495 00451 00490 00478 00551 00480 00515010 09125 09539 09460 09529 09510 09505 00912 00997 00968 01105 00962 01031020 09156 09588 09522 09521 09506 09498 01881 02113 02023 02209 01924 02062033 09201 09663 09620 09507 09499 09489 03311 04052 03718 03645 03174 03401050 09281 09788 09751 09510 09492 09500 05606 09797 07415 05528 04814 05159067 09372 09856 09812 09504 09500 09492 08470 17544 15776 07398 06442 06904

15

005 09244 09517 09443 09507 09506 09499 00366 00386 00377 00415 00380 00398010 09250 09523 09446 09494 09501 09475 00737 00781 00761 00830 00760 00796020 09294 09592 09537 09516 09509 09507 01520 01637 01584 01660 01521 01593033 09324 09681 09634 09502 09493 09489 02669 03016 02861 02737 02508 02626050 09418 09811 09783 09501 09490 09497 04495 05917 05267 04141 03794 03973067 09528 09894 09862 09496 09510 09493 06819 13361 10306 05564 05097 05338

25

005 09356 09513 09458 09504 09500 09505 00281 00290 00284 00302 00287 00295010 09338 09509 09452 09491 09481 09484 00566 00586 00573 00604 00574 00590020 09383 09580 09527 09497 09495 09491 01167 01219 01188 01209 01149 01181033 09453 09701 09664 09510 09497 09505 02043 02192 02118 01990 01892 01945050 09575 09839 09816 09520 09516 09512 03456 04004 03783 03024 02875 02956067 09651 09920 09899 09512 09503 09505 05205 07126 06382 04045 03845 03953

50

005 09400 09504 09458 09504 09488 09499 00198 00201 00197 00204 00199 00202010 09431 09520 09473 09493 09491 09492 00398 00405 00398 00409 00399 00405020 09479 09581 09534 09496 09491 09491 00819 00837 00821 00817 00797 00808033 09581 09695 09669 09506 09510 09502 01437 01490 01457 01349 01316 01334050 09686 09853 09834 09518 09512 09514 02420 02615 02538 02044 01994 02022067 09776 09940 09927 09507 09506 09510 03652 04272 04089 02740 02673 02710

100

005 09454 09502 09463 09496 09496 09492 00139 00140 00138 00141 00140 00141010 09479 09528 09494 09511 09502 09507 00281 00283 00279 00283 00280 00282020 09545 09590 09554 09500 09501 09501 00578 00584 00576 00566 00559 00563033 09621 09697 09675 09493 09489 09491 01013 01034 01018 00934 00923 00929050 09758 09844 09834 09489 09486 09488 01705 01789 01757 01416 01399 01408067 09849 09946 09939 09495 09489 09492 02570 02840 02775 01896 01873 01886

Thus the 100(1minus120572) confidence interval for 1205810based on the

pivotal quantity 119876 is

1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886 (27)

where 119886 119887 gt 0 119886 lt 119887 and the length of confidence intervalfor 1205810is defined as

119871 = 1205810radic119899(

1

radic119886minus

1

radic119887) (28)

In order to find the shortest-length confidence interval for 1205810

the following problem has to be solved

goal min119886119887

1205810radic119899(

1

radic119886minus

1

radic119887)

constraint int119887

119886

119891119876(119902) 119889119902 = 1 minus 120572

(29)

where 119891119876is the probability density function of central chi-

square distribution with 119899 degrees of freedom From Casella

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Probability and Statistics

interval but it is usually more accurate and nearly exactunder normality Panichkitkosolkul [19] modified McKayrsquosconfidence interval by replacing the sample coefficient ofvariation with the maximum likelihood estimator for anormal distribution Sharma and Krishna [20] introducedthe asymptotic distribution and confidence interval of thereciprocal of the coefficient of variation which does notrequire any assumptions about the population distribution tobe made Miller [21] discussed the approximate distributionof 120581 and proposed the approximate confidence interval for 120581in the case of a normal distributionTheperformance ofmanyconfidence intervals for 120581 obtained by McKayrsquos Millerrsquos andSharma-Krishnarsquos methods was compared under the samesimulation conditions by Ng [22]

Mahmoudvand and Hassani [23] proposed an approxi-mately unbiased estimator for 120581 in a normal distribution andalso used this estimator for constructing two approximateconfidence intervals for the coefficient of variation Theconfidence intervals for 120581 in normal and lognormal wereproposed by Koopmans et al [24] and Verrill [25] Buntaoand Niwitpong [26] also introduced an interval estimatingthe difference of the coefficient of variation for lognormaland delta-lognormal distributions Curto and Pinto [27] con-structed the confidence interval for 120581when random variablesare not independently and identically distributed Recentwork of Gulhar et al [28] has compared several confidenceintervals for estimating the population coefficient of variationbased on parametric nonparametric andmodifiedmethods

However the population mean may be known in severalphenomena The confidence intervals of the aforementionedauthors have not been used for estimating the populationcoefficient of variation for the normal distribution with aknown population mean Therefore our main aim in thispaper is to propose three confidence intervals for 120581 in anormal distribution with a known population mean

The organization of this paper is as follows In Section 2the theoretical background of the proposed confidence inter-vals is discussed The investigations of the performance ofthe proposed confidence interval through a Monte Carlosimulation study are presented in Section 3 A comparisonof the confidence intervals is also illustrated by using anempirical application in Section 4 Conclusions are providedin the final section

2 Theoretical Results

In this section the mean and variance of the estimator ofthe coefficient of variation in a normal distribution witha known population mean are considered In addition wewill introduce an unbiased estimator for the coefficient ofvariation obtain its variance and finally construct threeconfidence intervals normal approximation shortest-lengthand equal-tailed confidence intervals

If the population mean is known to be 1205830 then the

population coefficient of variation is given by 1205810= 120590120583

0 The

sample estimate of 1205810is

1205810=1198780

1205830

(2)

where 11987820= 119899minus1sum

119899

119894=1(119883119894minus 1205830)2 To find the expectation of (2)

we have to prove the following lemma

Lemma 1 Let 1198831 1198832 119883

119899be a random sample from

normal distribution with known mean 1205830and variance 1205902 and

let 11987820= 119899minus1sum

119899

119894=1(119883119894minus 1205830)2 Then

119864 (1198780) = 119888119899+1120590

V119886119903 (1198780) = (1 minus 119888

2

119899+1) 1205902

(3)

where 119888119899+1

= radic2119899(Γ((119899 + 1)2)Γ(1198992))

Proof of Lemma 1 By definition

1198782

0=1

119899

119899

sum119894=1

(119883119894minus 1205830)2=1205902

119899

119899

sum119894=1

1198852

119894 (4)

where 119885119894= (119883119894minus 1205830)120590 sim 119873(0 1)

Thus

11989911987820

1205902sim 1205942

119899 (5)

Let 1198782 = (119899 minus 1)minus1sum119899

119894=1(119883119894minus 119883)2 and 1198782

lowast=

119899minus1sum119899+1

119894=1(119883119894minus 119883)2 From Theorem B of Rice [29 page 197]

the distribution of (119899 minus 1)11987821205902 is central chi-square distribu-tion with 119899minus 1 degrees of freedom Similarly the distributionof 1198991198782lowast1205902 is central chi-square distribution with 119899 degrees of

freedom that is

1198991198782

lowast

1205902sim 1205942

119899 (6)

One can see that [30 page 181]

119864 (119878) = 119888119899120590 (7)

where 119888119899= radic2(119899 minus 1)(Γ(1198992)Γ((119899 minus 1)2))

Similarly

119864 (119878lowast) = 119888119899+1120590 (8)

where 119888119899+1

= radic2119899(Γ((119899 + 1)2)Γ(1198992))Equations (5) and (6) are equivalent Thus we obtain

119864(1198780) = 119864(119878

lowast) = 119888

119899+1120590 Next we will find the variance of

1198780

var (1198780) = 119864 (119878

2

0) minus [119864 (119878

0)]2= 1205902minus 1198882

119899+11205902= (1 minus 119888

2

119899+1) 1205902

(9)

By using Lemma 1 we can show that the mean andvariance of 120581

0are

119864 (1205810) =

119888119899+1

120590

1205830

= 119888119899+11205810 (10)

var (1205810) = (

1 minus 1198882119899+1

12058320

)1205902= (1 minus 119888

2

119899+1) 1205812

0 (11)

Journal of Probability and Statistics 3

Cov

erag

e pro

babi

litie

s

086

090

094

01 02 03 04 05 061205810

n = 5

MillerMcKayVangel

ApproxShortestEqual-tailed

(a)C

over

age p

roba

bilit

ies

01 02 03 04 05 061205810

096

092

088

n = 15

MillerMcKayVangel

ApproxShortestEqual-tailed

(b)

Cov

erag

e pro

babi

litie

s

01 02 03 04 05 061205810

098

096

094

092

090

n = 25

MillerMcKayVangel

ApproxShortestEqual-tailed

(c)

Cov

erag

e pro

babi

litie

s

01 02 03 04 05 061205810

098

096

094

092

090

n = 50

MillerMcKayVangel

ApproxShortestEqual-tailed

(d)

Figure 1 The estimated coverage probabilities of 90 confidence intervals for the coefficient of variation in a normal distribution with aknown population mean

Note that 119888119899+1

rarr 1 as 119899 rarr infin Therefore it follows that

lim119899rarrinfin

119864 (1205810) = 120581

0 (12)

It means that 1205810is asymptotically unbiased and asymptoti-

cally consistent for 1205810 From (10) the unbiased estimator of

1205810is

1205810=

1205810

119888119899+1

(13)

Using Lemma 1 the mean and variance of 1205810are given by

119864 (1205810) = 119864(

1205810

119888119899+1

) = 1205810 (14)

var (1205810) = var(

1205810

119888119899+1

) =1

1198882119899+1

var(1198780

1205830

)

=1

1198882119899+112058320

(1 minus 1198882

119899+1) 1205902= (

1 minus 1198882119899+1

1198882119899+1

)1205812

0

(15)

4 Journal of Probability and Statistics

Expe

cted

leng

ths

01 02 03 04 05 061205810

15

10

05

00

n = 5

MillerMcKayVangel

ApproxShortestEqual-tailed

(a)Ex

pect

ed le

ngth

s

01 02 03 04 05 061205810

00

02

04

06

08

n = 15

MillerMcKayVangel

ApproxShortestEqual-tailed

(b)

Expe

cted

leng

ths

01 02 03 04 05 061205810

05

04

03

02

01

00

n = 25

MillerMcKayVangel

ApproxShortestEqual-tailed

(c)

Expe

cted

leng

ths

01 02 03 04 05 061205810

030

020

010

000

n = 50

MillerMcKayVangel

ApproxShortestEqual-tailed

(d)

Figure 2The expected lengths of 90 confidence intervals for the coefficient of variation in a normal distribution with a known populationmean

Thus

lim119899rarrinfin

var (1205810) = 0 (16)

Hence 1205810is also asymptotically consistent for 120581

0 Next we

examine the accuracy of 1205810from another point view Let us

first consider the following theorem

Theorem 2 Let 1198831 1198832 119883

119899be a random sample from

a probability density function 119891(119909) which has unknownparameter 120579 If 120579 is an unbiased estimator of 120579 it can be shown

under very general conditions that the variance of 120579must satisfythe inequality

var (120579) ge 1

119899119864 (minus12059721205971205792 ln119891 (119909))=

1

119899119868 (120579) (17)

where 119868(120579) is the Fisher information This is known as theCramer-Rao inequality If var(120579) = 1(119899119868(120579)) the estimator120579 is said to be efficient

Proof of Theorem 2 See [31 pages 377ndash379]

Journal of Probability and Statistics 5

Weight

Freq

uenc

y

2500 3500 4500 5500

20

15

10

5

0

(a)

Weight

2000 3000 4000 5000 6000

Den

sity

6e minus 04

4e minus 04

2e minus 04

0e + 00

(b)

6000

5000

4000

3000

(c)

Theoretical quantiles

Sam

ple q

uant

iles

6000

5000

4000

3000

minus2 minus1 0 1 2

(d)

Figure 3 (a) Histogram (b) density plot (c) Box-and-Whisker plot and (d) normal quantile-quantile plot of the weights of 61 one-monthold infants

By setting 120579 = 1205810= 120590120583

0in Theorem 2 it is easy to show

that

var (0) ge

12058120

2119899 (18)

where 0is any unbiased estimator of 120581

0 This means that the

variance for the efficient estimator of 1205810is 120581202119899

From (15) we will show that (1minus1198882119899+1)1198882119899+1

rarr 1(2119899minus1)The asymptotic expansion of the gamma function ratio is [32]

Γ (119895 + (12))

Γ (119895)= radic119895(1 minus

1

8119895+

1

1281198952+ sdot sdot sdot ) (19)

Now if 119895 = 1198992 in (19) we have

119888119899+1

= radic2

119899

Γ ((119899 + 1) 2)

Γ (1198992)

= radic2

119899[radic

119899

2(1 minus

1

4119899+

1

321198992+ sdot sdot sdot )]

= 1 minus1

4119899+ o( 1

11989932)

(20)

Thus we obtain

1198882

119899+1= 1 minus

1

2119899+ o( 1

1198992)

1 minus 1198882119899+1

1198882119899+1

997888rarr1

2119899 minus 1

(21)

6 Journal of Probability and Statistics

Table 1 The values of 119886 and 119887 for the shortest-length confidence interval for 1205810

dfConfidence levels

090 095 099119886 119887 119886 119887 119886 119887

2 02065 125208 01015 151194 00200 2082643 05654 131532 03449 155897 01140 2098564 10200 141800 06918 165735 02937 2183715 15352 153498 11092 177432 05461 2298676 20930 165807 15776 189954 08567 2426187 26828 178391 20851 202863 12143 2560178 32981 191099 26235 215953 16107 2697499 39343 203848 31874 229118 20394 28364310 45883 216598 37729 242303 24958 29760211 52573 229325 43768 255476 29760 31158012 59397 242016 49967 268618 34771 32554313 66337 254666 56308 281717 39968 33947414 73382 267269 62776 294769 45329 35335815 80521 279825 69357 307770 50840 36719216 87745 292334 76042 320720 56487 38096817 95047 304796 82820 333619 62256 39468818 102421 317212 89685 346467 68139 40834719 109861 329585 96629 359266 74126 42195220 117362 341915 103647 372016 80209 43549821 124919 354205 110733 384720 86383 44898922 132530 366455 117882 397379 92640 46242623 140191 378668 125092 409995 98976 47581024 147899 390844 132357 422570 105385 48914425 155650 402986 139675 435105 111864 50242826 163443 415095 147043 447601 118408 51566527 171275 427171 154458 460060 125014 52885628 179144 439217 161917 472483 131678 54200229 187049 451234 169419 484872 138397 55510730 194987 463222 176961 497229 145170 56816940 275919 581755 254233 619217 215331 69680850 359012 698342 334085 738920 288879 82253460 443661 813479 415794 856914 364863 94606370 529501 927487 498923 973573 442711 106786780 616290 1040584 583183 1089153 522044 118827290 703860 1152925 668374 1203839 602597 1307514100 792086 1264628 754347 1317767 684177 1425771150 1240372 1816128 1192737 1879079 1103262 2006194200 1696646 2359748 1640642 2431025 1534834 2574375250 2158057 2898273 2094667 2976910 1974440 3134620300 2623132 3433155 2553057 3518461 2419776 3689185

Therefore var(1205810) rarr 1205812

0(2119899 minus 1) This means that 120581

0is

asymptotically efficient (see (18)) In the following sectionthree confidence intervals for 120581

0are proposed

21 Normal Approximation Confidence Interval Using thenormal approximate we have

119911 =1205810minus 1205810

var (1205810)=

1205810119888119899+1

minus 1205810

radic(1 minus 1198882119899+1) 120581201198882119899+1

=1205810minus 119888119899+11205810

1205810radic1 minus 1198882

119899+1

997888rarr 119873(0 1)

(22)

Journal of Probability and Statistics 7

Table 2 The estimated coverage probabilities and expected lengths of 90 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08499 09066 08858 09016 09023 08979 00555 00607 00582 00741 00587 00675010 08518 09069 08866 09024 09008 08988 01120 01237 01177 01482 01174 01349020 08524 09130 08960 09036 08990 09006 02315 02689 02457 02963 02347 02696033 08572 09258 09136 09038 08999 09001 04099 05872 04528 04895 03878 04453050 08664 09430 09321 09036 08994 09001 06959 12123 09360 07409 05869 06741067 08773 09578 09428 09031 09000 08992 10603 15764 16394 09947 07880 09050

10

005 08747 09031 08870 09020 08996 09006 00379 00396 00382 00431 00388 00416010 08792 09052 08899 09024 09001 09014 00765 00804 00773 00864 00778 00833020 08802 09135 09002 09013 08993 09001 01576 01686 01603 01726 01553 01664033 08899 09304 09202 09017 09021 09015 02778 03140 02893 02853 02566 02750050 08999 09527 09451 09007 09004 08995 04709 06575 05323 04329 03895 04174067 09129 09694 09600 09018 08999 08992 07128 14205 10257 05801 05218 05593

15

005 08846 09010 08870 09000 08989 08988 00307 00316 00306 00333 00311 00326010 08866 09065 08925 09011 09013 09001 00618 00638 00617 00666 00622 00652020 08913 09127 09007 09006 08993 08987 01271 01328 01275 01330 01242 01301033 09046 09308 09218 09012 09022 09013 02239 02418 02286 02200 02054 02151050 09150 09544 09477 09000 09004 08991 03787 04522 04087 03338 03116 03264067 09280 09725 09661 09010 09002 08999 05713 08900 07010 04466 04170 04367

25

005 08933 09056 08932 09029 09035 09021 00236 00240 00233 00247 00238 00244010 08948 09054 08939 09010 09014 09004 00475 00485 00471 00495 00475 00489020 09022 09146 09042 09028 09008 09021 00977 01003 00971 00988 00949 00976033 09144 09318 09238 09005 09016 09004 01716 01796 01727 01630 01566 01610050 09285 09548 09491 08976 08977 08978 02893 03207 03027 02471 02374 02440067 09430 09768 09722 09018 08999 09008 04363 05418 04930 03310 03179 03268

50

005 08941 08992 08905 08993 08977 08989 00166 00167 00164 00170 00166 00168010 08996 09043 08949 09004 09007 08997 00334 00337 00330 00339 00333 00337020 09061 09118 09041 08994 08989 08996 00688 00697 00680 00678 00665 00674033 09220 09314 09253 08997 08996 08994 01206 01236 01204 01118 01096 01112050 09436 09583 09539 09009 09022 09010 02031 02153 02084 01695 01662 01685067 09588 09801 09770 09010 09009 09008 03062 03460 03309 02271 02226 02257

100

005 08998 09026 08961 09019 08902 09017 00117 00117 00115 00118 00113 00118010 09000 09031 08959 08995 08878 08992 00236 00237 00233 00236 00227 00236020 09110 09131 09072 09011 08901 09008 00485 00488 00480 00472 00453 00471033 09277 09329 09282 09015 08900 09012 00850 00863 00847 00779 00748 00777050 09485 09589 09561 09001 08885 08998 01429 01486 01455 01180 01133 01177067 09678 09810 09790 09020 08910 09021 02157 02347 02289 01582 01519 01578

Therefore the 100(1 minus 120572) confidence interval for 1205810based

on (22) is

1205810

119888119899+1

+ 1199111minus1205722

radic1 minus 1198882119899+1

le 1205810le

1205810

119888119899+1

minus 1199111minus1205722

radic1 minus 1198882119899+1

(23)

where 1199111minus1205722

is the 100(1 minus 1205722) percentile of the standardnormal distribution

22 Shortest-Length Confidence Interval A pivotal quantityfor 1205902 is

119876 =11989911987820

1205902sim 1205942

119899 (24)

Converting the statement

119875(119886 le11989911987820

1205902le 119887) = 1 minus 120572 (25)

we can write

119875(1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886) = 1 minus 120572 (26)

8 Journal of Probability and Statistics

Table 3 The estimated coverage probabilities and expected lengths of 95 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08829 09533 09440 09538 09504 09511 00661 00785 00762 01058 00758 00870010 08827 09537 09457 09549 09501 09506 01333 01608 01544 02113 01513 01737020 08847 09578 09508 09548 09500 09507 02756 03630 03282 04226 03026 03475033 08904 09647 09599 09542 09501 09501 04880 08954 06498 06986 05001 05743050 08934 09711 09656 09537 09487 09491 08276 13796 14333 10561 07561 08683067 09042 09795 09721 09548 09495 09502 12550 15758 19791 14140 10124 11625

10

005 09115 09522 09440 09511 09502 09495 00451 00490 00478 00551 00480 00515010 09125 09539 09460 09529 09510 09505 00912 00997 00968 01105 00962 01031020 09156 09588 09522 09521 09506 09498 01881 02113 02023 02209 01924 02062033 09201 09663 09620 09507 09499 09489 03311 04052 03718 03645 03174 03401050 09281 09788 09751 09510 09492 09500 05606 09797 07415 05528 04814 05159067 09372 09856 09812 09504 09500 09492 08470 17544 15776 07398 06442 06904

15

005 09244 09517 09443 09507 09506 09499 00366 00386 00377 00415 00380 00398010 09250 09523 09446 09494 09501 09475 00737 00781 00761 00830 00760 00796020 09294 09592 09537 09516 09509 09507 01520 01637 01584 01660 01521 01593033 09324 09681 09634 09502 09493 09489 02669 03016 02861 02737 02508 02626050 09418 09811 09783 09501 09490 09497 04495 05917 05267 04141 03794 03973067 09528 09894 09862 09496 09510 09493 06819 13361 10306 05564 05097 05338

25

005 09356 09513 09458 09504 09500 09505 00281 00290 00284 00302 00287 00295010 09338 09509 09452 09491 09481 09484 00566 00586 00573 00604 00574 00590020 09383 09580 09527 09497 09495 09491 01167 01219 01188 01209 01149 01181033 09453 09701 09664 09510 09497 09505 02043 02192 02118 01990 01892 01945050 09575 09839 09816 09520 09516 09512 03456 04004 03783 03024 02875 02956067 09651 09920 09899 09512 09503 09505 05205 07126 06382 04045 03845 03953

50

005 09400 09504 09458 09504 09488 09499 00198 00201 00197 00204 00199 00202010 09431 09520 09473 09493 09491 09492 00398 00405 00398 00409 00399 00405020 09479 09581 09534 09496 09491 09491 00819 00837 00821 00817 00797 00808033 09581 09695 09669 09506 09510 09502 01437 01490 01457 01349 01316 01334050 09686 09853 09834 09518 09512 09514 02420 02615 02538 02044 01994 02022067 09776 09940 09927 09507 09506 09510 03652 04272 04089 02740 02673 02710

100

005 09454 09502 09463 09496 09496 09492 00139 00140 00138 00141 00140 00141010 09479 09528 09494 09511 09502 09507 00281 00283 00279 00283 00280 00282020 09545 09590 09554 09500 09501 09501 00578 00584 00576 00566 00559 00563033 09621 09697 09675 09493 09489 09491 01013 01034 01018 00934 00923 00929050 09758 09844 09834 09489 09486 09488 01705 01789 01757 01416 01399 01408067 09849 09946 09939 09495 09489 09492 02570 02840 02775 01896 01873 01886

Thus the 100(1minus120572) confidence interval for 1205810based on the

pivotal quantity 119876 is

1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886 (27)

where 119886 119887 gt 0 119886 lt 119887 and the length of confidence intervalfor 1205810is defined as

119871 = 1205810radic119899(

1

radic119886minus

1

radic119887) (28)

In order to find the shortest-length confidence interval for 1205810

the following problem has to be solved

goal min119886119887

1205810radic119899(

1

radic119886minus

1

radic119887)

constraint int119887

119886

119891119876(119902) 119889119902 = 1 minus 120572

(29)

where 119891119876is the probability density function of central chi-

square distribution with 119899 degrees of freedom From Casella

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 3

Cov

erag

e pro

babi

litie

s

086

090

094

01 02 03 04 05 061205810

n = 5

MillerMcKayVangel

ApproxShortestEqual-tailed

(a)C

over

age p

roba

bilit

ies

01 02 03 04 05 061205810

096

092

088

n = 15

MillerMcKayVangel

ApproxShortestEqual-tailed

(b)

Cov

erag

e pro

babi

litie

s

01 02 03 04 05 061205810

098

096

094

092

090

n = 25

MillerMcKayVangel

ApproxShortestEqual-tailed

(c)

Cov

erag

e pro

babi

litie

s

01 02 03 04 05 061205810

098

096

094

092

090

n = 50

MillerMcKayVangel

ApproxShortestEqual-tailed

(d)

Figure 1 The estimated coverage probabilities of 90 confidence intervals for the coefficient of variation in a normal distribution with aknown population mean

Note that 119888119899+1

rarr 1 as 119899 rarr infin Therefore it follows that

lim119899rarrinfin

119864 (1205810) = 120581

0 (12)

It means that 1205810is asymptotically unbiased and asymptoti-

cally consistent for 1205810 From (10) the unbiased estimator of

1205810is

1205810=

1205810

119888119899+1

(13)

Using Lemma 1 the mean and variance of 1205810are given by

119864 (1205810) = 119864(

1205810

119888119899+1

) = 1205810 (14)

var (1205810) = var(

1205810

119888119899+1

) =1

1198882119899+1

var(1198780

1205830

)

=1

1198882119899+112058320

(1 minus 1198882

119899+1) 1205902= (

1 minus 1198882119899+1

1198882119899+1

)1205812

0

(15)

4 Journal of Probability and Statistics

Expe

cted

leng

ths

01 02 03 04 05 061205810

15

10

05

00

n = 5

MillerMcKayVangel

ApproxShortestEqual-tailed

(a)Ex

pect

ed le

ngth

s

01 02 03 04 05 061205810

00

02

04

06

08

n = 15

MillerMcKayVangel

ApproxShortestEqual-tailed

(b)

Expe

cted

leng

ths

01 02 03 04 05 061205810

05

04

03

02

01

00

n = 25

MillerMcKayVangel

ApproxShortestEqual-tailed

(c)

Expe

cted

leng

ths

01 02 03 04 05 061205810

030

020

010

000

n = 50

MillerMcKayVangel

ApproxShortestEqual-tailed

(d)

Figure 2The expected lengths of 90 confidence intervals for the coefficient of variation in a normal distribution with a known populationmean

Thus

lim119899rarrinfin

var (1205810) = 0 (16)

Hence 1205810is also asymptotically consistent for 120581

0 Next we

examine the accuracy of 1205810from another point view Let us

first consider the following theorem

Theorem 2 Let 1198831 1198832 119883

119899be a random sample from

a probability density function 119891(119909) which has unknownparameter 120579 If 120579 is an unbiased estimator of 120579 it can be shown

under very general conditions that the variance of 120579must satisfythe inequality

var (120579) ge 1

119899119864 (minus12059721205971205792 ln119891 (119909))=

1

119899119868 (120579) (17)

where 119868(120579) is the Fisher information This is known as theCramer-Rao inequality If var(120579) = 1(119899119868(120579)) the estimator120579 is said to be efficient

Proof of Theorem 2 See [31 pages 377ndash379]

Journal of Probability and Statistics 5

Weight

Freq

uenc

y

2500 3500 4500 5500

20

15

10

5

0

(a)

Weight

2000 3000 4000 5000 6000

Den

sity

6e minus 04

4e minus 04

2e minus 04

0e + 00

(b)

6000

5000

4000

3000

(c)

Theoretical quantiles

Sam

ple q

uant

iles

6000

5000

4000

3000

minus2 minus1 0 1 2

(d)

Figure 3 (a) Histogram (b) density plot (c) Box-and-Whisker plot and (d) normal quantile-quantile plot of the weights of 61 one-monthold infants

By setting 120579 = 1205810= 120590120583

0in Theorem 2 it is easy to show

that

var (0) ge

12058120

2119899 (18)

where 0is any unbiased estimator of 120581

0 This means that the

variance for the efficient estimator of 1205810is 120581202119899

From (15) we will show that (1minus1198882119899+1)1198882119899+1

rarr 1(2119899minus1)The asymptotic expansion of the gamma function ratio is [32]

Γ (119895 + (12))

Γ (119895)= radic119895(1 minus

1

8119895+

1

1281198952+ sdot sdot sdot ) (19)

Now if 119895 = 1198992 in (19) we have

119888119899+1

= radic2

119899

Γ ((119899 + 1) 2)

Γ (1198992)

= radic2

119899[radic

119899

2(1 minus

1

4119899+

1

321198992+ sdot sdot sdot )]

= 1 minus1

4119899+ o( 1

11989932)

(20)

Thus we obtain

1198882

119899+1= 1 minus

1

2119899+ o( 1

1198992)

1 minus 1198882119899+1

1198882119899+1

997888rarr1

2119899 minus 1

(21)

6 Journal of Probability and Statistics

Table 1 The values of 119886 and 119887 for the shortest-length confidence interval for 1205810

dfConfidence levels

090 095 099119886 119887 119886 119887 119886 119887

2 02065 125208 01015 151194 00200 2082643 05654 131532 03449 155897 01140 2098564 10200 141800 06918 165735 02937 2183715 15352 153498 11092 177432 05461 2298676 20930 165807 15776 189954 08567 2426187 26828 178391 20851 202863 12143 2560178 32981 191099 26235 215953 16107 2697499 39343 203848 31874 229118 20394 28364310 45883 216598 37729 242303 24958 29760211 52573 229325 43768 255476 29760 31158012 59397 242016 49967 268618 34771 32554313 66337 254666 56308 281717 39968 33947414 73382 267269 62776 294769 45329 35335815 80521 279825 69357 307770 50840 36719216 87745 292334 76042 320720 56487 38096817 95047 304796 82820 333619 62256 39468818 102421 317212 89685 346467 68139 40834719 109861 329585 96629 359266 74126 42195220 117362 341915 103647 372016 80209 43549821 124919 354205 110733 384720 86383 44898922 132530 366455 117882 397379 92640 46242623 140191 378668 125092 409995 98976 47581024 147899 390844 132357 422570 105385 48914425 155650 402986 139675 435105 111864 50242826 163443 415095 147043 447601 118408 51566527 171275 427171 154458 460060 125014 52885628 179144 439217 161917 472483 131678 54200229 187049 451234 169419 484872 138397 55510730 194987 463222 176961 497229 145170 56816940 275919 581755 254233 619217 215331 69680850 359012 698342 334085 738920 288879 82253460 443661 813479 415794 856914 364863 94606370 529501 927487 498923 973573 442711 106786780 616290 1040584 583183 1089153 522044 118827290 703860 1152925 668374 1203839 602597 1307514100 792086 1264628 754347 1317767 684177 1425771150 1240372 1816128 1192737 1879079 1103262 2006194200 1696646 2359748 1640642 2431025 1534834 2574375250 2158057 2898273 2094667 2976910 1974440 3134620300 2623132 3433155 2553057 3518461 2419776 3689185

Therefore var(1205810) rarr 1205812

0(2119899 minus 1) This means that 120581

0is

asymptotically efficient (see (18)) In the following sectionthree confidence intervals for 120581

0are proposed

21 Normal Approximation Confidence Interval Using thenormal approximate we have

119911 =1205810minus 1205810

var (1205810)=

1205810119888119899+1

minus 1205810

radic(1 minus 1198882119899+1) 120581201198882119899+1

=1205810minus 119888119899+11205810

1205810radic1 minus 1198882

119899+1

997888rarr 119873(0 1)

(22)

Journal of Probability and Statistics 7

Table 2 The estimated coverage probabilities and expected lengths of 90 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08499 09066 08858 09016 09023 08979 00555 00607 00582 00741 00587 00675010 08518 09069 08866 09024 09008 08988 01120 01237 01177 01482 01174 01349020 08524 09130 08960 09036 08990 09006 02315 02689 02457 02963 02347 02696033 08572 09258 09136 09038 08999 09001 04099 05872 04528 04895 03878 04453050 08664 09430 09321 09036 08994 09001 06959 12123 09360 07409 05869 06741067 08773 09578 09428 09031 09000 08992 10603 15764 16394 09947 07880 09050

10

005 08747 09031 08870 09020 08996 09006 00379 00396 00382 00431 00388 00416010 08792 09052 08899 09024 09001 09014 00765 00804 00773 00864 00778 00833020 08802 09135 09002 09013 08993 09001 01576 01686 01603 01726 01553 01664033 08899 09304 09202 09017 09021 09015 02778 03140 02893 02853 02566 02750050 08999 09527 09451 09007 09004 08995 04709 06575 05323 04329 03895 04174067 09129 09694 09600 09018 08999 08992 07128 14205 10257 05801 05218 05593

15

005 08846 09010 08870 09000 08989 08988 00307 00316 00306 00333 00311 00326010 08866 09065 08925 09011 09013 09001 00618 00638 00617 00666 00622 00652020 08913 09127 09007 09006 08993 08987 01271 01328 01275 01330 01242 01301033 09046 09308 09218 09012 09022 09013 02239 02418 02286 02200 02054 02151050 09150 09544 09477 09000 09004 08991 03787 04522 04087 03338 03116 03264067 09280 09725 09661 09010 09002 08999 05713 08900 07010 04466 04170 04367

25

005 08933 09056 08932 09029 09035 09021 00236 00240 00233 00247 00238 00244010 08948 09054 08939 09010 09014 09004 00475 00485 00471 00495 00475 00489020 09022 09146 09042 09028 09008 09021 00977 01003 00971 00988 00949 00976033 09144 09318 09238 09005 09016 09004 01716 01796 01727 01630 01566 01610050 09285 09548 09491 08976 08977 08978 02893 03207 03027 02471 02374 02440067 09430 09768 09722 09018 08999 09008 04363 05418 04930 03310 03179 03268

50

005 08941 08992 08905 08993 08977 08989 00166 00167 00164 00170 00166 00168010 08996 09043 08949 09004 09007 08997 00334 00337 00330 00339 00333 00337020 09061 09118 09041 08994 08989 08996 00688 00697 00680 00678 00665 00674033 09220 09314 09253 08997 08996 08994 01206 01236 01204 01118 01096 01112050 09436 09583 09539 09009 09022 09010 02031 02153 02084 01695 01662 01685067 09588 09801 09770 09010 09009 09008 03062 03460 03309 02271 02226 02257

100

005 08998 09026 08961 09019 08902 09017 00117 00117 00115 00118 00113 00118010 09000 09031 08959 08995 08878 08992 00236 00237 00233 00236 00227 00236020 09110 09131 09072 09011 08901 09008 00485 00488 00480 00472 00453 00471033 09277 09329 09282 09015 08900 09012 00850 00863 00847 00779 00748 00777050 09485 09589 09561 09001 08885 08998 01429 01486 01455 01180 01133 01177067 09678 09810 09790 09020 08910 09021 02157 02347 02289 01582 01519 01578

Therefore the 100(1 minus 120572) confidence interval for 1205810based

on (22) is

1205810

119888119899+1

+ 1199111minus1205722

radic1 minus 1198882119899+1

le 1205810le

1205810

119888119899+1

minus 1199111minus1205722

radic1 minus 1198882119899+1

(23)

where 1199111minus1205722

is the 100(1 minus 1205722) percentile of the standardnormal distribution

22 Shortest-Length Confidence Interval A pivotal quantityfor 1205902 is

119876 =11989911987820

1205902sim 1205942

119899 (24)

Converting the statement

119875(119886 le11989911987820

1205902le 119887) = 1 minus 120572 (25)

we can write

119875(1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886) = 1 minus 120572 (26)

8 Journal of Probability and Statistics

Table 3 The estimated coverage probabilities and expected lengths of 95 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08829 09533 09440 09538 09504 09511 00661 00785 00762 01058 00758 00870010 08827 09537 09457 09549 09501 09506 01333 01608 01544 02113 01513 01737020 08847 09578 09508 09548 09500 09507 02756 03630 03282 04226 03026 03475033 08904 09647 09599 09542 09501 09501 04880 08954 06498 06986 05001 05743050 08934 09711 09656 09537 09487 09491 08276 13796 14333 10561 07561 08683067 09042 09795 09721 09548 09495 09502 12550 15758 19791 14140 10124 11625

10

005 09115 09522 09440 09511 09502 09495 00451 00490 00478 00551 00480 00515010 09125 09539 09460 09529 09510 09505 00912 00997 00968 01105 00962 01031020 09156 09588 09522 09521 09506 09498 01881 02113 02023 02209 01924 02062033 09201 09663 09620 09507 09499 09489 03311 04052 03718 03645 03174 03401050 09281 09788 09751 09510 09492 09500 05606 09797 07415 05528 04814 05159067 09372 09856 09812 09504 09500 09492 08470 17544 15776 07398 06442 06904

15

005 09244 09517 09443 09507 09506 09499 00366 00386 00377 00415 00380 00398010 09250 09523 09446 09494 09501 09475 00737 00781 00761 00830 00760 00796020 09294 09592 09537 09516 09509 09507 01520 01637 01584 01660 01521 01593033 09324 09681 09634 09502 09493 09489 02669 03016 02861 02737 02508 02626050 09418 09811 09783 09501 09490 09497 04495 05917 05267 04141 03794 03973067 09528 09894 09862 09496 09510 09493 06819 13361 10306 05564 05097 05338

25

005 09356 09513 09458 09504 09500 09505 00281 00290 00284 00302 00287 00295010 09338 09509 09452 09491 09481 09484 00566 00586 00573 00604 00574 00590020 09383 09580 09527 09497 09495 09491 01167 01219 01188 01209 01149 01181033 09453 09701 09664 09510 09497 09505 02043 02192 02118 01990 01892 01945050 09575 09839 09816 09520 09516 09512 03456 04004 03783 03024 02875 02956067 09651 09920 09899 09512 09503 09505 05205 07126 06382 04045 03845 03953

50

005 09400 09504 09458 09504 09488 09499 00198 00201 00197 00204 00199 00202010 09431 09520 09473 09493 09491 09492 00398 00405 00398 00409 00399 00405020 09479 09581 09534 09496 09491 09491 00819 00837 00821 00817 00797 00808033 09581 09695 09669 09506 09510 09502 01437 01490 01457 01349 01316 01334050 09686 09853 09834 09518 09512 09514 02420 02615 02538 02044 01994 02022067 09776 09940 09927 09507 09506 09510 03652 04272 04089 02740 02673 02710

100

005 09454 09502 09463 09496 09496 09492 00139 00140 00138 00141 00140 00141010 09479 09528 09494 09511 09502 09507 00281 00283 00279 00283 00280 00282020 09545 09590 09554 09500 09501 09501 00578 00584 00576 00566 00559 00563033 09621 09697 09675 09493 09489 09491 01013 01034 01018 00934 00923 00929050 09758 09844 09834 09489 09486 09488 01705 01789 01757 01416 01399 01408067 09849 09946 09939 09495 09489 09492 02570 02840 02775 01896 01873 01886

Thus the 100(1minus120572) confidence interval for 1205810based on the

pivotal quantity 119876 is

1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886 (27)

where 119886 119887 gt 0 119886 lt 119887 and the length of confidence intervalfor 1205810is defined as

119871 = 1205810radic119899(

1

radic119886minus

1

radic119887) (28)

In order to find the shortest-length confidence interval for 1205810

the following problem has to be solved

goal min119886119887

1205810radic119899(

1

radic119886minus

1

radic119887)

constraint int119887

119886

119891119876(119902) 119889119902 = 1 minus 120572

(29)

where 119891119876is the probability density function of central chi-

square distribution with 119899 degrees of freedom From Casella

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Probability and Statistics

Expe

cted

leng

ths

01 02 03 04 05 061205810

15

10

05

00

n = 5

MillerMcKayVangel

ApproxShortestEqual-tailed

(a)Ex

pect

ed le

ngth

s

01 02 03 04 05 061205810

00

02

04

06

08

n = 15

MillerMcKayVangel

ApproxShortestEqual-tailed

(b)

Expe

cted

leng

ths

01 02 03 04 05 061205810

05

04

03

02

01

00

n = 25

MillerMcKayVangel

ApproxShortestEqual-tailed

(c)

Expe

cted

leng

ths

01 02 03 04 05 061205810

030

020

010

000

n = 50

MillerMcKayVangel

ApproxShortestEqual-tailed

(d)

Figure 2The expected lengths of 90 confidence intervals for the coefficient of variation in a normal distribution with a known populationmean

Thus

lim119899rarrinfin

var (1205810) = 0 (16)

Hence 1205810is also asymptotically consistent for 120581

0 Next we

examine the accuracy of 1205810from another point view Let us

first consider the following theorem

Theorem 2 Let 1198831 1198832 119883

119899be a random sample from

a probability density function 119891(119909) which has unknownparameter 120579 If 120579 is an unbiased estimator of 120579 it can be shown

under very general conditions that the variance of 120579must satisfythe inequality

var (120579) ge 1

119899119864 (minus12059721205971205792 ln119891 (119909))=

1

119899119868 (120579) (17)

where 119868(120579) is the Fisher information This is known as theCramer-Rao inequality If var(120579) = 1(119899119868(120579)) the estimator120579 is said to be efficient

Proof of Theorem 2 See [31 pages 377ndash379]

Journal of Probability and Statistics 5

Weight

Freq

uenc

y

2500 3500 4500 5500

20

15

10

5

0

(a)

Weight

2000 3000 4000 5000 6000

Den

sity

6e minus 04

4e minus 04

2e minus 04

0e + 00

(b)

6000

5000

4000

3000

(c)

Theoretical quantiles

Sam

ple q

uant

iles

6000

5000

4000

3000

minus2 minus1 0 1 2

(d)

Figure 3 (a) Histogram (b) density plot (c) Box-and-Whisker plot and (d) normal quantile-quantile plot of the weights of 61 one-monthold infants

By setting 120579 = 1205810= 120590120583

0in Theorem 2 it is easy to show

that

var (0) ge

12058120

2119899 (18)

where 0is any unbiased estimator of 120581

0 This means that the

variance for the efficient estimator of 1205810is 120581202119899

From (15) we will show that (1minus1198882119899+1)1198882119899+1

rarr 1(2119899minus1)The asymptotic expansion of the gamma function ratio is [32]

Γ (119895 + (12))

Γ (119895)= radic119895(1 minus

1

8119895+

1

1281198952+ sdot sdot sdot ) (19)

Now if 119895 = 1198992 in (19) we have

119888119899+1

= radic2

119899

Γ ((119899 + 1) 2)

Γ (1198992)

= radic2

119899[radic

119899

2(1 minus

1

4119899+

1

321198992+ sdot sdot sdot )]

= 1 minus1

4119899+ o( 1

11989932)

(20)

Thus we obtain

1198882

119899+1= 1 minus

1

2119899+ o( 1

1198992)

1 minus 1198882119899+1

1198882119899+1

997888rarr1

2119899 minus 1

(21)

6 Journal of Probability and Statistics

Table 1 The values of 119886 and 119887 for the shortest-length confidence interval for 1205810

dfConfidence levels

090 095 099119886 119887 119886 119887 119886 119887

2 02065 125208 01015 151194 00200 2082643 05654 131532 03449 155897 01140 2098564 10200 141800 06918 165735 02937 2183715 15352 153498 11092 177432 05461 2298676 20930 165807 15776 189954 08567 2426187 26828 178391 20851 202863 12143 2560178 32981 191099 26235 215953 16107 2697499 39343 203848 31874 229118 20394 28364310 45883 216598 37729 242303 24958 29760211 52573 229325 43768 255476 29760 31158012 59397 242016 49967 268618 34771 32554313 66337 254666 56308 281717 39968 33947414 73382 267269 62776 294769 45329 35335815 80521 279825 69357 307770 50840 36719216 87745 292334 76042 320720 56487 38096817 95047 304796 82820 333619 62256 39468818 102421 317212 89685 346467 68139 40834719 109861 329585 96629 359266 74126 42195220 117362 341915 103647 372016 80209 43549821 124919 354205 110733 384720 86383 44898922 132530 366455 117882 397379 92640 46242623 140191 378668 125092 409995 98976 47581024 147899 390844 132357 422570 105385 48914425 155650 402986 139675 435105 111864 50242826 163443 415095 147043 447601 118408 51566527 171275 427171 154458 460060 125014 52885628 179144 439217 161917 472483 131678 54200229 187049 451234 169419 484872 138397 55510730 194987 463222 176961 497229 145170 56816940 275919 581755 254233 619217 215331 69680850 359012 698342 334085 738920 288879 82253460 443661 813479 415794 856914 364863 94606370 529501 927487 498923 973573 442711 106786780 616290 1040584 583183 1089153 522044 118827290 703860 1152925 668374 1203839 602597 1307514100 792086 1264628 754347 1317767 684177 1425771150 1240372 1816128 1192737 1879079 1103262 2006194200 1696646 2359748 1640642 2431025 1534834 2574375250 2158057 2898273 2094667 2976910 1974440 3134620300 2623132 3433155 2553057 3518461 2419776 3689185

Therefore var(1205810) rarr 1205812

0(2119899 minus 1) This means that 120581

0is

asymptotically efficient (see (18)) In the following sectionthree confidence intervals for 120581

0are proposed

21 Normal Approximation Confidence Interval Using thenormal approximate we have

119911 =1205810minus 1205810

var (1205810)=

1205810119888119899+1

minus 1205810

radic(1 minus 1198882119899+1) 120581201198882119899+1

=1205810minus 119888119899+11205810

1205810radic1 minus 1198882

119899+1

997888rarr 119873(0 1)

(22)

Journal of Probability and Statistics 7

Table 2 The estimated coverage probabilities and expected lengths of 90 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08499 09066 08858 09016 09023 08979 00555 00607 00582 00741 00587 00675010 08518 09069 08866 09024 09008 08988 01120 01237 01177 01482 01174 01349020 08524 09130 08960 09036 08990 09006 02315 02689 02457 02963 02347 02696033 08572 09258 09136 09038 08999 09001 04099 05872 04528 04895 03878 04453050 08664 09430 09321 09036 08994 09001 06959 12123 09360 07409 05869 06741067 08773 09578 09428 09031 09000 08992 10603 15764 16394 09947 07880 09050

10

005 08747 09031 08870 09020 08996 09006 00379 00396 00382 00431 00388 00416010 08792 09052 08899 09024 09001 09014 00765 00804 00773 00864 00778 00833020 08802 09135 09002 09013 08993 09001 01576 01686 01603 01726 01553 01664033 08899 09304 09202 09017 09021 09015 02778 03140 02893 02853 02566 02750050 08999 09527 09451 09007 09004 08995 04709 06575 05323 04329 03895 04174067 09129 09694 09600 09018 08999 08992 07128 14205 10257 05801 05218 05593

15

005 08846 09010 08870 09000 08989 08988 00307 00316 00306 00333 00311 00326010 08866 09065 08925 09011 09013 09001 00618 00638 00617 00666 00622 00652020 08913 09127 09007 09006 08993 08987 01271 01328 01275 01330 01242 01301033 09046 09308 09218 09012 09022 09013 02239 02418 02286 02200 02054 02151050 09150 09544 09477 09000 09004 08991 03787 04522 04087 03338 03116 03264067 09280 09725 09661 09010 09002 08999 05713 08900 07010 04466 04170 04367

25

005 08933 09056 08932 09029 09035 09021 00236 00240 00233 00247 00238 00244010 08948 09054 08939 09010 09014 09004 00475 00485 00471 00495 00475 00489020 09022 09146 09042 09028 09008 09021 00977 01003 00971 00988 00949 00976033 09144 09318 09238 09005 09016 09004 01716 01796 01727 01630 01566 01610050 09285 09548 09491 08976 08977 08978 02893 03207 03027 02471 02374 02440067 09430 09768 09722 09018 08999 09008 04363 05418 04930 03310 03179 03268

50

005 08941 08992 08905 08993 08977 08989 00166 00167 00164 00170 00166 00168010 08996 09043 08949 09004 09007 08997 00334 00337 00330 00339 00333 00337020 09061 09118 09041 08994 08989 08996 00688 00697 00680 00678 00665 00674033 09220 09314 09253 08997 08996 08994 01206 01236 01204 01118 01096 01112050 09436 09583 09539 09009 09022 09010 02031 02153 02084 01695 01662 01685067 09588 09801 09770 09010 09009 09008 03062 03460 03309 02271 02226 02257

100

005 08998 09026 08961 09019 08902 09017 00117 00117 00115 00118 00113 00118010 09000 09031 08959 08995 08878 08992 00236 00237 00233 00236 00227 00236020 09110 09131 09072 09011 08901 09008 00485 00488 00480 00472 00453 00471033 09277 09329 09282 09015 08900 09012 00850 00863 00847 00779 00748 00777050 09485 09589 09561 09001 08885 08998 01429 01486 01455 01180 01133 01177067 09678 09810 09790 09020 08910 09021 02157 02347 02289 01582 01519 01578

Therefore the 100(1 minus 120572) confidence interval for 1205810based

on (22) is

1205810

119888119899+1

+ 1199111minus1205722

radic1 minus 1198882119899+1

le 1205810le

1205810

119888119899+1

minus 1199111minus1205722

radic1 minus 1198882119899+1

(23)

where 1199111minus1205722

is the 100(1 minus 1205722) percentile of the standardnormal distribution

22 Shortest-Length Confidence Interval A pivotal quantityfor 1205902 is

119876 =11989911987820

1205902sim 1205942

119899 (24)

Converting the statement

119875(119886 le11989911987820

1205902le 119887) = 1 minus 120572 (25)

we can write

119875(1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886) = 1 minus 120572 (26)

8 Journal of Probability and Statistics

Table 3 The estimated coverage probabilities and expected lengths of 95 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08829 09533 09440 09538 09504 09511 00661 00785 00762 01058 00758 00870010 08827 09537 09457 09549 09501 09506 01333 01608 01544 02113 01513 01737020 08847 09578 09508 09548 09500 09507 02756 03630 03282 04226 03026 03475033 08904 09647 09599 09542 09501 09501 04880 08954 06498 06986 05001 05743050 08934 09711 09656 09537 09487 09491 08276 13796 14333 10561 07561 08683067 09042 09795 09721 09548 09495 09502 12550 15758 19791 14140 10124 11625

10

005 09115 09522 09440 09511 09502 09495 00451 00490 00478 00551 00480 00515010 09125 09539 09460 09529 09510 09505 00912 00997 00968 01105 00962 01031020 09156 09588 09522 09521 09506 09498 01881 02113 02023 02209 01924 02062033 09201 09663 09620 09507 09499 09489 03311 04052 03718 03645 03174 03401050 09281 09788 09751 09510 09492 09500 05606 09797 07415 05528 04814 05159067 09372 09856 09812 09504 09500 09492 08470 17544 15776 07398 06442 06904

15

005 09244 09517 09443 09507 09506 09499 00366 00386 00377 00415 00380 00398010 09250 09523 09446 09494 09501 09475 00737 00781 00761 00830 00760 00796020 09294 09592 09537 09516 09509 09507 01520 01637 01584 01660 01521 01593033 09324 09681 09634 09502 09493 09489 02669 03016 02861 02737 02508 02626050 09418 09811 09783 09501 09490 09497 04495 05917 05267 04141 03794 03973067 09528 09894 09862 09496 09510 09493 06819 13361 10306 05564 05097 05338

25

005 09356 09513 09458 09504 09500 09505 00281 00290 00284 00302 00287 00295010 09338 09509 09452 09491 09481 09484 00566 00586 00573 00604 00574 00590020 09383 09580 09527 09497 09495 09491 01167 01219 01188 01209 01149 01181033 09453 09701 09664 09510 09497 09505 02043 02192 02118 01990 01892 01945050 09575 09839 09816 09520 09516 09512 03456 04004 03783 03024 02875 02956067 09651 09920 09899 09512 09503 09505 05205 07126 06382 04045 03845 03953

50

005 09400 09504 09458 09504 09488 09499 00198 00201 00197 00204 00199 00202010 09431 09520 09473 09493 09491 09492 00398 00405 00398 00409 00399 00405020 09479 09581 09534 09496 09491 09491 00819 00837 00821 00817 00797 00808033 09581 09695 09669 09506 09510 09502 01437 01490 01457 01349 01316 01334050 09686 09853 09834 09518 09512 09514 02420 02615 02538 02044 01994 02022067 09776 09940 09927 09507 09506 09510 03652 04272 04089 02740 02673 02710

100

005 09454 09502 09463 09496 09496 09492 00139 00140 00138 00141 00140 00141010 09479 09528 09494 09511 09502 09507 00281 00283 00279 00283 00280 00282020 09545 09590 09554 09500 09501 09501 00578 00584 00576 00566 00559 00563033 09621 09697 09675 09493 09489 09491 01013 01034 01018 00934 00923 00929050 09758 09844 09834 09489 09486 09488 01705 01789 01757 01416 01399 01408067 09849 09946 09939 09495 09489 09492 02570 02840 02775 01896 01873 01886

Thus the 100(1minus120572) confidence interval for 1205810based on the

pivotal quantity 119876 is

1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886 (27)

where 119886 119887 gt 0 119886 lt 119887 and the length of confidence intervalfor 1205810is defined as

119871 = 1205810radic119899(

1

radic119886minus

1

radic119887) (28)

In order to find the shortest-length confidence interval for 1205810

the following problem has to be solved

goal min119886119887

1205810radic119899(

1

radic119886minus

1

radic119887)

constraint int119887

119886

119891119876(119902) 119889119902 = 1 minus 120572

(29)

where 119891119876is the probability density function of central chi-

square distribution with 119899 degrees of freedom From Casella

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 5

Weight

Freq

uenc

y

2500 3500 4500 5500

20

15

10

5

0

(a)

Weight

2000 3000 4000 5000 6000

Den

sity

6e minus 04

4e minus 04

2e minus 04

0e + 00

(b)

6000

5000

4000

3000

(c)

Theoretical quantiles

Sam

ple q

uant

iles

6000

5000

4000

3000

minus2 minus1 0 1 2

(d)

Figure 3 (a) Histogram (b) density plot (c) Box-and-Whisker plot and (d) normal quantile-quantile plot of the weights of 61 one-monthold infants

By setting 120579 = 1205810= 120590120583

0in Theorem 2 it is easy to show

that

var (0) ge

12058120

2119899 (18)

where 0is any unbiased estimator of 120581

0 This means that the

variance for the efficient estimator of 1205810is 120581202119899

From (15) we will show that (1minus1198882119899+1)1198882119899+1

rarr 1(2119899minus1)The asymptotic expansion of the gamma function ratio is [32]

Γ (119895 + (12))

Γ (119895)= radic119895(1 minus

1

8119895+

1

1281198952+ sdot sdot sdot ) (19)

Now if 119895 = 1198992 in (19) we have

119888119899+1

= radic2

119899

Γ ((119899 + 1) 2)

Γ (1198992)

= radic2

119899[radic

119899

2(1 minus

1

4119899+

1

321198992+ sdot sdot sdot )]

= 1 minus1

4119899+ o( 1

11989932)

(20)

Thus we obtain

1198882

119899+1= 1 minus

1

2119899+ o( 1

1198992)

1 minus 1198882119899+1

1198882119899+1

997888rarr1

2119899 minus 1

(21)

6 Journal of Probability and Statistics

Table 1 The values of 119886 and 119887 for the shortest-length confidence interval for 1205810

dfConfidence levels

090 095 099119886 119887 119886 119887 119886 119887

2 02065 125208 01015 151194 00200 2082643 05654 131532 03449 155897 01140 2098564 10200 141800 06918 165735 02937 2183715 15352 153498 11092 177432 05461 2298676 20930 165807 15776 189954 08567 2426187 26828 178391 20851 202863 12143 2560178 32981 191099 26235 215953 16107 2697499 39343 203848 31874 229118 20394 28364310 45883 216598 37729 242303 24958 29760211 52573 229325 43768 255476 29760 31158012 59397 242016 49967 268618 34771 32554313 66337 254666 56308 281717 39968 33947414 73382 267269 62776 294769 45329 35335815 80521 279825 69357 307770 50840 36719216 87745 292334 76042 320720 56487 38096817 95047 304796 82820 333619 62256 39468818 102421 317212 89685 346467 68139 40834719 109861 329585 96629 359266 74126 42195220 117362 341915 103647 372016 80209 43549821 124919 354205 110733 384720 86383 44898922 132530 366455 117882 397379 92640 46242623 140191 378668 125092 409995 98976 47581024 147899 390844 132357 422570 105385 48914425 155650 402986 139675 435105 111864 50242826 163443 415095 147043 447601 118408 51566527 171275 427171 154458 460060 125014 52885628 179144 439217 161917 472483 131678 54200229 187049 451234 169419 484872 138397 55510730 194987 463222 176961 497229 145170 56816940 275919 581755 254233 619217 215331 69680850 359012 698342 334085 738920 288879 82253460 443661 813479 415794 856914 364863 94606370 529501 927487 498923 973573 442711 106786780 616290 1040584 583183 1089153 522044 118827290 703860 1152925 668374 1203839 602597 1307514100 792086 1264628 754347 1317767 684177 1425771150 1240372 1816128 1192737 1879079 1103262 2006194200 1696646 2359748 1640642 2431025 1534834 2574375250 2158057 2898273 2094667 2976910 1974440 3134620300 2623132 3433155 2553057 3518461 2419776 3689185

Therefore var(1205810) rarr 1205812

0(2119899 minus 1) This means that 120581

0is

asymptotically efficient (see (18)) In the following sectionthree confidence intervals for 120581

0are proposed

21 Normal Approximation Confidence Interval Using thenormal approximate we have

119911 =1205810minus 1205810

var (1205810)=

1205810119888119899+1

minus 1205810

radic(1 minus 1198882119899+1) 120581201198882119899+1

=1205810minus 119888119899+11205810

1205810radic1 minus 1198882

119899+1

997888rarr 119873(0 1)

(22)

Journal of Probability and Statistics 7

Table 2 The estimated coverage probabilities and expected lengths of 90 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08499 09066 08858 09016 09023 08979 00555 00607 00582 00741 00587 00675010 08518 09069 08866 09024 09008 08988 01120 01237 01177 01482 01174 01349020 08524 09130 08960 09036 08990 09006 02315 02689 02457 02963 02347 02696033 08572 09258 09136 09038 08999 09001 04099 05872 04528 04895 03878 04453050 08664 09430 09321 09036 08994 09001 06959 12123 09360 07409 05869 06741067 08773 09578 09428 09031 09000 08992 10603 15764 16394 09947 07880 09050

10

005 08747 09031 08870 09020 08996 09006 00379 00396 00382 00431 00388 00416010 08792 09052 08899 09024 09001 09014 00765 00804 00773 00864 00778 00833020 08802 09135 09002 09013 08993 09001 01576 01686 01603 01726 01553 01664033 08899 09304 09202 09017 09021 09015 02778 03140 02893 02853 02566 02750050 08999 09527 09451 09007 09004 08995 04709 06575 05323 04329 03895 04174067 09129 09694 09600 09018 08999 08992 07128 14205 10257 05801 05218 05593

15

005 08846 09010 08870 09000 08989 08988 00307 00316 00306 00333 00311 00326010 08866 09065 08925 09011 09013 09001 00618 00638 00617 00666 00622 00652020 08913 09127 09007 09006 08993 08987 01271 01328 01275 01330 01242 01301033 09046 09308 09218 09012 09022 09013 02239 02418 02286 02200 02054 02151050 09150 09544 09477 09000 09004 08991 03787 04522 04087 03338 03116 03264067 09280 09725 09661 09010 09002 08999 05713 08900 07010 04466 04170 04367

25

005 08933 09056 08932 09029 09035 09021 00236 00240 00233 00247 00238 00244010 08948 09054 08939 09010 09014 09004 00475 00485 00471 00495 00475 00489020 09022 09146 09042 09028 09008 09021 00977 01003 00971 00988 00949 00976033 09144 09318 09238 09005 09016 09004 01716 01796 01727 01630 01566 01610050 09285 09548 09491 08976 08977 08978 02893 03207 03027 02471 02374 02440067 09430 09768 09722 09018 08999 09008 04363 05418 04930 03310 03179 03268

50

005 08941 08992 08905 08993 08977 08989 00166 00167 00164 00170 00166 00168010 08996 09043 08949 09004 09007 08997 00334 00337 00330 00339 00333 00337020 09061 09118 09041 08994 08989 08996 00688 00697 00680 00678 00665 00674033 09220 09314 09253 08997 08996 08994 01206 01236 01204 01118 01096 01112050 09436 09583 09539 09009 09022 09010 02031 02153 02084 01695 01662 01685067 09588 09801 09770 09010 09009 09008 03062 03460 03309 02271 02226 02257

100

005 08998 09026 08961 09019 08902 09017 00117 00117 00115 00118 00113 00118010 09000 09031 08959 08995 08878 08992 00236 00237 00233 00236 00227 00236020 09110 09131 09072 09011 08901 09008 00485 00488 00480 00472 00453 00471033 09277 09329 09282 09015 08900 09012 00850 00863 00847 00779 00748 00777050 09485 09589 09561 09001 08885 08998 01429 01486 01455 01180 01133 01177067 09678 09810 09790 09020 08910 09021 02157 02347 02289 01582 01519 01578

Therefore the 100(1 minus 120572) confidence interval for 1205810based

on (22) is

1205810

119888119899+1

+ 1199111minus1205722

radic1 minus 1198882119899+1

le 1205810le

1205810

119888119899+1

minus 1199111minus1205722

radic1 minus 1198882119899+1

(23)

where 1199111minus1205722

is the 100(1 minus 1205722) percentile of the standardnormal distribution

22 Shortest-Length Confidence Interval A pivotal quantityfor 1205902 is

119876 =11989911987820

1205902sim 1205942

119899 (24)

Converting the statement

119875(119886 le11989911987820

1205902le 119887) = 1 minus 120572 (25)

we can write

119875(1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886) = 1 minus 120572 (26)

8 Journal of Probability and Statistics

Table 3 The estimated coverage probabilities and expected lengths of 95 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08829 09533 09440 09538 09504 09511 00661 00785 00762 01058 00758 00870010 08827 09537 09457 09549 09501 09506 01333 01608 01544 02113 01513 01737020 08847 09578 09508 09548 09500 09507 02756 03630 03282 04226 03026 03475033 08904 09647 09599 09542 09501 09501 04880 08954 06498 06986 05001 05743050 08934 09711 09656 09537 09487 09491 08276 13796 14333 10561 07561 08683067 09042 09795 09721 09548 09495 09502 12550 15758 19791 14140 10124 11625

10

005 09115 09522 09440 09511 09502 09495 00451 00490 00478 00551 00480 00515010 09125 09539 09460 09529 09510 09505 00912 00997 00968 01105 00962 01031020 09156 09588 09522 09521 09506 09498 01881 02113 02023 02209 01924 02062033 09201 09663 09620 09507 09499 09489 03311 04052 03718 03645 03174 03401050 09281 09788 09751 09510 09492 09500 05606 09797 07415 05528 04814 05159067 09372 09856 09812 09504 09500 09492 08470 17544 15776 07398 06442 06904

15

005 09244 09517 09443 09507 09506 09499 00366 00386 00377 00415 00380 00398010 09250 09523 09446 09494 09501 09475 00737 00781 00761 00830 00760 00796020 09294 09592 09537 09516 09509 09507 01520 01637 01584 01660 01521 01593033 09324 09681 09634 09502 09493 09489 02669 03016 02861 02737 02508 02626050 09418 09811 09783 09501 09490 09497 04495 05917 05267 04141 03794 03973067 09528 09894 09862 09496 09510 09493 06819 13361 10306 05564 05097 05338

25

005 09356 09513 09458 09504 09500 09505 00281 00290 00284 00302 00287 00295010 09338 09509 09452 09491 09481 09484 00566 00586 00573 00604 00574 00590020 09383 09580 09527 09497 09495 09491 01167 01219 01188 01209 01149 01181033 09453 09701 09664 09510 09497 09505 02043 02192 02118 01990 01892 01945050 09575 09839 09816 09520 09516 09512 03456 04004 03783 03024 02875 02956067 09651 09920 09899 09512 09503 09505 05205 07126 06382 04045 03845 03953

50

005 09400 09504 09458 09504 09488 09499 00198 00201 00197 00204 00199 00202010 09431 09520 09473 09493 09491 09492 00398 00405 00398 00409 00399 00405020 09479 09581 09534 09496 09491 09491 00819 00837 00821 00817 00797 00808033 09581 09695 09669 09506 09510 09502 01437 01490 01457 01349 01316 01334050 09686 09853 09834 09518 09512 09514 02420 02615 02538 02044 01994 02022067 09776 09940 09927 09507 09506 09510 03652 04272 04089 02740 02673 02710

100

005 09454 09502 09463 09496 09496 09492 00139 00140 00138 00141 00140 00141010 09479 09528 09494 09511 09502 09507 00281 00283 00279 00283 00280 00282020 09545 09590 09554 09500 09501 09501 00578 00584 00576 00566 00559 00563033 09621 09697 09675 09493 09489 09491 01013 01034 01018 00934 00923 00929050 09758 09844 09834 09489 09486 09488 01705 01789 01757 01416 01399 01408067 09849 09946 09939 09495 09489 09492 02570 02840 02775 01896 01873 01886

Thus the 100(1minus120572) confidence interval for 1205810based on the

pivotal quantity 119876 is

1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886 (27)

where 119886 119887 gt 0 119886 lt 119887 and the length of confidence intervalfor 1205810is defined as

119871 = 1205810radic119899(

1

radic119886minus

1

radic119887) (28)

In order to find the shortest-length confidence interval for 1205810

the following problem has to be solved

goal min119886119887

1205810radic119899(

1

radic119886minus

1

radic119887)

constraint int119887

119886

119891119876(119902) 119889119902 = 1 minus 120572

(29)

where 119891119876is the probability density function of central chi-

square distribution with 119899 degrees of freedom From Casella

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Probability and Statistics

Table 1 The values of 119886 and 119887 for the shortest-length confidence interval for 1205810

dfConfidence levels

090 095 099119886 119887 119886 119887 119886 119887

2 02065 125208 01015 151194 00200 2082643 05654 131532 03449 155897 01140 2098564 10200 141800 06918 165735 02937 2183715 15352 153498 11092 177432 05461 2298676 20930 165807 15776 189954 08567 2426187 26828 178391 20851 202863 12143 2560178 32981 191099 26235 215953 16107 2697499 39343 203848 31874 229118 20394 28364310 45883 216598 37729 242303 24958 29760211 52573 229325 43768 255476 29760 31158012 59397 242016 49967 268618 34771 32554313 66337 254666 56308 281717 39968 33947414 73382 267269 62776 294769 45329 35335815 80521 279825 69357 307770 50840 36719216 87745 292334 76042 320720 56487 38096817 95047 304796 82820 333619 62256 39468818 102421 317212 89685 346467 68139 40834719 109861 329585 96629 359266 74126 42195220 117362 341915 103647 372016 80209 43549821 124919 354205 110733 384720 86383 44898922 132530 366455 117882 397379 92640 46242623 140191 378668 125092 409995 98976 47581024 147899 390844 132357 422570 105385 48914425 155650 402986 139675 435105 111864 50242826 163443 415095 147043 447601 118408 51566527 171275 427171 154458 460060 125014 52885628 179144 439217 161917 472483 131678 54200229 187049 451234 169419 484872 138397 55510730 194987 463222 176961 497229 145170 56816940 275919 581755 254233 619217 215331 69680850 359012 698342 334085 738920 288879 82253460 443661 813479 415794 856914 364863 94606370 529501 927487 498923 973573 442711 106786780 616290 1040584 583183 1089153 522044 118827290 703860 1152925 668374 1203839 602597 1307514100 792086 1264628 754347 1317767 684177 1425771150 1240372 1816128 1192737 1879079 1103262 2006194200 1696646 2359748 1640642 2431025 1534834 2574375250 2158057 2898273 2094667 2976910 1974440 3134620300 2623132 3433155 2553057 3518461 2419776 3689185

Therefore var(1205810) rarr 1205812

0(2119899 minus 1) This means that 120581

0is

asymptotically efficient (see (18)) In the following sectionthree confidence intervals for 120581

0are proposed

21 Normal Approximation Confidence Interval Using thenormal approximate we have

119911 =1205810minus 1205810

var (1205810)=

1205810119888119899+1

minus 1205810

radic(1 minus 1198882119899+1) 120581201198882119899+1

=1205810minus 119888119899+11205810

1205810radic1 minus 1198882

119899+1

997888rarr 119873(0 1)

(22)

Journal of Probability and Statistics 7

Table 2 The estimated coverage probabilities and expected lengths of 90 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08499 09066 08858 09016 09023 08979 00555 00607 00582 00741 00587 00675010 08518 09069 08866 09024 09008 08988 01120 01237 01177 01482 01174 01349020 08524 09130 08960 09036 08990 09006 02315 02689 02457 02963 02347 02696033 08572 09258 09136 09038 08999 09001 04099 05872 04528 04895 03878 04453050 08664 09430 09321 09036 08994 09001 06959 12123 09360 07409 05869 06741067 08773 09578 09428 09031 09000 08992 10603 15764 16394 09947 07880 09050

10

005 08747 09031 08870 09020 08996 09006 00379 00396 00382 00431 00388 00416010 08792 09052 08899 09024 09001 09014 00765 00804 00773 00864 00778 00833020 08802 09135 09002 09013 08993 09001 01576 01686 01603 01726 01553 01664033 08899 09304 09202 09017 09021 09015 02778 03140 02893 02853 02566 02750050 08999 09527 09451 09007 09004 08995 04709 06575 05323 04329 03895 04174067 09129 09694 09600 09018 08999 08992 07128 14205 10257 05801 05218 05593

15

005 08846 09010 08870 09000 08989 08988 00307 00316 00306 00333 00311 00326010 08866 09065 08925 09011 09013 09001 00618 00638 00617 00666 00622 00652020 08913 09127 09007 09006 08993 08987 01271 01328 01275 01330 01242 01301033 09046 09308 09218 09012 09022 09013 02239 02418 02286 02200 02054 02151050 09150 09544 09477 09000 09004 08991 03787 04522 04087 03338 03116 03264067 09280 09725 09661 09010 09002 08999 05713 08900 07010 04466 04170 04367

25

005 08933 09056 08932 09029 09035 09021 00236 00240 00233 00247 00238 00244010 08948 09054 08939 09010 09014 09004 00475 00485 00471 00495 00475 00489020 09022 09146 09042 09028 09008 09021 00977 01003 00971 00988 00949 00976033 09144 09318 09238 09005 09016 09004 01716 01796 01727 01630 01566 01610050 09285 09548 09491 08976 08977 08978 02893 03207 03027 02471 02374 02440067 09430 09768 09722 09018 08999 09008 04363 05418 04930 03310 03179 03268

50

005 08941 08992 08905 08993 08977 08989 00166 00167 00164 00170 00166 00168010 08996 09043 08949 09004 09007 08997 00334 00337 00330 00339 00333 00337020 09061 09118 09041 08994 08989 08996 00688 00697 00680 00678 00665 00674033 09220 09314 09253 08997 08996 08994 01206 01236 01204 01118 01096 01112050 09436 09583 09539 09009 09022 09010 02031 02153 02084 01695 01662 01685067 09588 09801 09770 09010 09009 09008 03062 03460 03309 02271 02226 02257

100

005 08998 09026 08961 09019 08902 09017 00117 00117 00115 00118 00113 00118010 09000 09031 08959 08995 08878 08992 00236 00237 00233 00236 00227 00236020 09110 09131 09072 09011 08901 09008 00485 00488 00480 00472 00453 00471033 09277 09329 09282 09015 08900 09012 00850 00863 00847 00779 00748 00777050 09485 09589 09561 09001 08885 08998 01429 01486 01455 01180 01133 01177067 09678 09810 09790 09020 08910 09021 02157 02347 02289 01582 01519 01578

Therefore the 100(1 minus 120572) confidence interval for 1205810based

on (22) is

1205810

119888119899+1

+ 1199111minus1205722

radic1 minus 1198882119899+1

le 1205810le

1205810

119888119899+1

minus 1199111minus1205722

radic1 minus 1198882119899+1

(23)

where 1199111minus1205722

is the 100(1 minus 1205722) percentile of the standardnormal distribution

22 Shortest-Length Confidence Interval A pivotal quantityfor 1205902 is

119876 =11989911987820

1205902sim 1205942

119899 (24)

Converting the statement

119875(119886 le11989911987820

1205902le 119887) = 1 minus 120572 (25)

we can write

119875(1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886) = 1 minus 120572 (26)

8 Journal of Probability and Statistics

Table 3 The estimated coverage probabilities and expected lengths of 95 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08829 09533 09440 09538 09504 09511 00661 00785 00762 01058 00758 00870010 08827 09537 09457 09549 09501 09506 01333 01608 01544 02113 01513 01737020 08847 09578 09508 09548 09500 09507 02756 03630 03282 04226 03026 03475033 08904 09647 09599 09542 09501 09501 04880 08954 06498 06986 05001 05743050 08934 09711 09656 09537 09487 09491 08276 13796 14333 10561 07561 08683067 09042 09795 09721 09548 09495 09502 12550 15758 19791 14140 10124 11625

10

005 09115 09522 09440 09511 09502 09495 00451 00490 00478 00551 00480 00515010 09125 09539 09460 09529 09510 09505 00912 00997 00968 01105 00962 01031020 09156 09588 09522 09521 09506 09498 01881 02113 02023 02209 01924 02062033 09201 09663 09620 09507 09499 09489 03311 04052 03718 03645 03174 03401050 09281 09788 09751 09510 09492 09500 05606 09797 07415 05528 04814 05159067 09372 09856 09812 09504 09500 09492 08470 17544 15776 07398 06442 06904

15

005 09244 09517 09443 09507 09506 09499 00366 00386 00377 00415 00380 00398010 09250 09523 09446 09494 09501 09475 00737 00781 00761 00830 00760 00796020 09294 09592 09537 09516 09509 09507 01520 01637 01584 01660 01521 01593033 09324 09681 09634 09502 09493 09489 02669 03016 02861 02737 02508 02626050 09418 09811 09783 09501 09490 09497 04495 05917 05267 04141 03794 03973067 09528 09894 09862 09496 09510 09493 06819 13361 10306 05564 05097 05338

25

005 09356 09513 09458 09504 09500 09505 00281 00290 00284 00302 00287 00295010 09338 09509 09452 09491 09481 09484 00566 00586 00573 00604 00574 00590020 09383 09580 09527 09497 09495 09491 01167 01219 01188 01209 01149 01181033 09453 09701 09664 09510 09497 09505 02043 02192 02118 01990 01892 01945050 09575 09839 09816 09520 09516 09512 03456 04004 03783 03024 02875 02956067 09651 09920 09899 09512 09503 09505 05205 07126 06382 04045 03845 03953

50

005 09400 09504 09458 09504 09488 09499 00198 00201 00197 00204 00199 00202010 09431 09520 09473 09493 09491 09492 00398 00405 00398 00409 00399 00405020 09479 09581 09534 09496 09491 09491 00819 00837 00821 00817 00797 00808033 09581 09695 09669 09506 09510 09502 01437 01490 01457 01349 01316 01334050 09686 09853 09834 09518 09512 09514 02420 02615 02538 02044 01994 02022067 09776 09940 09927 09507 09506 09510 03652 04272 04089 02740 02673 02710

100

005 09454 09502 09463 09496 09496 09492 00139 00140 00138 00141 00140 00141010 09479 09528 09494 09511 09502 09507 00281 00283 00279 00283 00280 00282020 09545 09590 09554 09500 09501 09501 00578 00584 00576 00566 00559 00563033 09621 09697 09675 09493 09489 09491 01013 01034 01018 00934 00923 00929050 09758 09844 09834 09489 09486 09488 01705 01789 01757 01416 01399 01408067 09849 09946 09939 09495 09489 09492 02570 02840 02775 01896 01873 01886

Thus the 100(1minus120572) confidence interval for 1205810based on the

pivotal quantity 119876 is

1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886 (27)

where 119886 119887 gt 0 119886 lt 119887 and the length of confidence intervalfor 1205810is defined as

119871 = 1205810radic119899(

1

radic119886minus

1

radic119887) (28)

In order to find the shortest-length confidence interval for 1205810

the following problem has to be solved

goal min119886119887

1205810radic119899(

1

radic119886minus

1

radic119887)

constraint int119887

119886

119891119876(119902) 119889119902 = 1 minus 120572

(29)

where 119891119876is the probability density function of central chi-

square distribution with 119899 degrees of freedom From Casella

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 7

Table 2 The estimated coverage probabilities and expected lengths of 90 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08499 09066 08858 09016 09023 08979 00555 00607 00582 00741 00587 00675010 08518 09069 08866 09024 09008 08988 01120 01237 01177 01482 01174 01349020 08524 09130 08960 09036 08990 09006 02315 02689 02457 02963 02347 02696033 08572 09258 09136 09038 08999 09001 04099 05872 04528 04895 03878 04453050 08664 09430 09321 09036 08994 09001 06959 12123 09360 07409 05869 06741067 08773 09578 09428 09031 09000 08992 10603 15764 16394 09947 07880 09050

10

005 08747 09031 08870 09020 08996 09006 00379 00396 00382 00431 00388 00416010 08792 09052 08899 09024 09001 09014 00765 00804 00773 00864 00778 00833020 08802 09135 09002 09013 08993 09001 01576 01686 01603 01726 01553 01664033 08899 09304 09202 09017 09021 09015 02778 03140 02893 02853 02566 02750050 08999 09527 09451 09007 09004 08995 04709 06575 05323 04329 03895 04174067 09129 09694 09600 09018 08999 08992 07128 14205 10257 05801 05218 05593

15

005 08846 09010 08870 09000 08989 08988 00307 00316 00306 00333 00311 00326010 08866 09065 08925 09011 09013 09001 00618 00638 00617 00666 00622 00652020 08913 09127 09007 09006 08993 08987 01271 01328 01275 01330 01242 01301033 09046 09308 09218 09012 09022 09013 02239 02418 02286 02200 02054 02151050 09150 09544 09477 09000 09004 08991 03787 04522 04087 03338 03116 03264067 09280 09725 09661 09010 09002 08999 05713 08900 07010 04466 04170 04367

25

005 08933 09056 08932 09029 09035 09021 00236 00240 00233 00247 00238 00244010 08948 09054 08939 09010 09014 09004 00475 00485 00471 00495 00475 00489020 09022 09146 09042 09028 09008 09021 00977 01003 00971 00988 00949 00976033 09144 09318 09238 09005 09016 09004 01716 01796 01727 01630 01566 01610050 09285 09548 09491 08976 08977 08978 02893 03207 03027 02471 02374 02440067 09430 09768 09722 09018 08999 09008 04363 05418 04930 03310 03179 03268

50

005 08941 08992 08905 08993 08977 08989 00166 00167 00164 00170 00166 00168010 08996 09043 08949 09004 09007 08997 00334 00337 00330 00339 00333 00337020 09061 09118 09041 08994 08989 08996 00688 00697 00680 00678 00665 00674033 09220 09314 09253 08997 08996 08994 01206 01236 01204 01118 01096 01112050 09436 09583 09539 09009 09022 09010 02031 02153 02084 01695 01662 01685067 09588 09801 09770 09010 09009 09008 03062 03460 03309 02271 02226 02257

100

005 08998 09026 08961 09019 08902 09017 00117 00117 00115 00118 00113 00118010 09000 09031 08959 08995 08878 08992 00236 00237 00233 00236 00227 00236020 09110 09131 09072 09011 08901 09008 00485 00488 00480 00472 00453 00471033 09277 09329 09282 09015 08900 09012 00850 00863 00847 00779 00748 00777050 09485 09589 09561 09001 08885 08998 01429 01486 01455 01180 01133 01177067 09678 09810 09790 09020 08910 09021 02157 02347 02289 01582 01519 01578

Therefore the 100(1 minus 120572) confidence interval for 1205810based

on (22) is

1205810

119888119899+1

+ 1199111minus1205722

radic1 minus 1198882119899+1

le 1205810le

1205810

119888119899+1

minus 1199111minus1205722

radic1 minus 1198882119899+1

(23)

where 1199111minus1205722

is the 100(1 minus 1205722) percentile of the standardnormal distribution

22 Shortest-Length Confidence Interval A pivotal quantityfor 1205902 is

119876 =11989911987820

1205902sim 1205942

119899 (24)

Converting the statement

119875(119886 le11989911987820

1205902le 119887) = 1 minus 120572 (25)

we can write

119875(1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886) = 1 minus 120572 (26)

8 Journal of Probability and Statistics

Table 3 The estimated coverage probabilities and expected lengths of 95 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08829 09533 09440 09538 09504 09511 00661 00785 00762 01058 00758 00870010 08827 09537 09457 09549 09501 09506 01333 01608 01544 02113 01513 01737020 08847 09578 09508 09548 09500 09507 02756 03630 03282 04226 03026 03475033 08904 09647 09599 09542 09501 09501 04880 08954 06498 06986 05001 05743050 08934 09711 09656 09537 09487 09491 08276 13796 14333 10561 07561 08683067 09042 09795 09721 09548 09495 09502 12550 15758 19791 14140 10124 11625

10

005 09115 09522 09440 09511 09502 09495 00451 00490 00478 00551 00480 00515010 09125 09539 09460 09529 09510 09505 00912 00997 00968 01105 00962 01031020 09156 09588 09522 09521 09506 09498 01881 02113 02023 02209 01924 02062033 09201 09663 09620 09507 09499 09489 03311 04052 03718 03645 03174 03401050 09281 09788 09751 09510 09492 09500 05606 09797 07415 05528 04814 05159067 09372 09856 09812 09504 09500 09492 08470 17544 15776 07398 06442 06904

15

005 09244 09517 09443 09507 09506 09499 00366 00386 00377 00415 00380 00398010 09250 09523 09446 09494 09501 09475 00737 00781 00761 00830 00760 00796020 09294 09592 09537 09516 09509 09507 01520 01637 01584 01660 01521 01593033 09324 09681 09634 09502 09493 09489 02669 03016 02861 02737 02508 02626050 09418 09811 09783 09501 09490 09497 04495 05917 05267 04141 03794 03973067 09528 09894 09862 09496 09510 09493 06819 13361 10306 05564 05097 05338

25

005 09356 09513 09458 09504 09500 09505 00281 00290 00284 00302 00287 00295010 09338 09509 09452 09491 09481 09484 00566 00586 00573 00604 00574 00590020 09383 09580 09527 09497 09495 09491 01167 01219 01188 01209 01149 01181033 09453 09701 09664 09510 09497 09505 02043 02192 02118 01990 01892 01945050 09575 09839 09816 09520 09516 09512 03456 04004 03783 03024 02875 02956067 09651 09920 09899 09512 09503 09505 05205 07126 06382 04045 03845 03953

50

005 09400 09504 09458 09504 09488 09499 00198 00201 00197 00204 00199 00202010 09431 09520 09473 09493 09491 09492 00398 00405 00398 00409 00399 00405020 09479 09581 09534 09496 09491 09491 00819 00837 00821 00817 00797 00808033 09581 09695 09669 09506 09510 09502 01437 01490 01457 01349 01316 01334050 09686 09853 09834 09518 09512 09514 02420 02615 02538 02044 01994 02022067 09776 09940 09927 09507 09506 09510 03652 04272 04089 02740 02673 02710

100

005 09454 09502 09463 09496 09496 09492 00139 00140 00138 00141 00140 00141010 09479 09528 09494 09511 09502 09507 00281 00283 00279 00283 00280 00282020 09545 09590 09554 09500 09501 09501 00578 00584 00576 00566 00559 00563033 09621 09697 09675 09493 09489 09491 01013 01034 01018 00934 00923 00929050 09758 09844 09834 09489 09486 09488 01705 01789 01757 01416 01399 01408067 09849 09946 09939 09495 09489 09492 02570 02840 02775 01896 01873 01886

Thus the 100(1minus120572) confidence interval for 1205810based on the

pivotal quantity 119876 is

1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886 (27)

where 119886 119887 gt 0 119886 lt 119887 and the length of confidence intervalfor 1205810is defined as

119871 = 1205810radic119899(

1

radic119886minus

1

radic119887) (28)

In order to find the shortest-length confidence interval for 1205810

the following problem has to be solved

goal min119886119887

1205810radic119899(

1

radic119886minus

1

radic119887)

constraint int119887

119886

119891119876(119902) 119889119902 = 1 minus 120572

(29)

where 119891119876is the probability density function of central chi-

square distribution with 119899 degrees of freedom From Casella

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Probability and Statistics

Table 3 The estimated coverage probabilities and expected lengths of 95 confidence intervals for the coefficient of variation in a normaldistribution with a known population mean

119899 1205810

Coverage probabilities Expected lengthsMiller McKay Vangel Approx Shortest Equal-tailed Miller McKay Vangel Approx Shortest Equal-tailed

5

005 08829 09533 09440 09538 09504 09511 00661 00785 00762 01058 00758 00870010 08827 09537 09457 09549 09501 09506 01333 01608 01544 02113 01513 01737020 08847 09578 09508 09548 09500 09507 02756 03630 03282 04226 03026 03475033 08904 09647 09599 09542 09501 09501 04880 08954 06498 06986 05001 05743050 08934 09711 09656 09537 09487 09491 08276 13796 14333 10561 07561 08683067 09042 09795 09721 09548 09495 09502 12550 15758 19791 14140 10124 11625

10

005 09115 09522 09440 09511 09502 09495 00451 00490 00478 00551 00480 00515010 09125 09539 09460 09529 09510 09505 00912 00997 00968 01105 00962 01031020 09156 09588 09522 09521 09506 09498 01881 02113 02023 02209 01924 02062033 09201 09663 09620 09507 09499 09489 03311 04052 03718 03645 03174 03401050 09281 09788 09751 09510 09492 09500 05606 09797 07415 05528 04814 05159067 09372 09856 09812 09504 09500 09492 08470 17544 15776 07398 06442 06904

15

005 09244 09517 09443 09507 09506 09499 00366 00386 00377 00415 00380 00398010 09250 09523 09446 09494 09501 09475 00737 00781 00761 00830 00760 00796020 09294 09592 09537 09516 09509 09507 01520 01637 01584 01660 01521 01593033 09324 09681 09634 09502 09493 09489 02669 03016 02861 02737 02508 02626050 09418 09811 09783 09501 09490 09497 04495 05917 05267 04141 03794 03973067 09528 09894 09862 09496 09510 09493 06819 13361 10306 05564 05097 05338

25

005 09356 09513 09458 09504 09500 09505 00281 00290 00284 00302 00287 00295010 09338 09509 09452 09491 09481 09484 00566 00586 00573 00604 00574 00590020 09383 09580 09527 09497 09495 09491 01167 01219 01188 01209 01149 01181033 09453 09701 09664 09510 09497 09505 02043 02192 02118 01990 01892 01945050 09575 09839 09816 09520 09516 09512 03456 04004 03783 03024 02875 02956067 09651 09920 09899 09512 09503 09505 05205 07126 06382 04045 03845 03953

50

005 09400 09504 09458 09504 09488 09499 00198 00201 00197 00204 00199 00202010 09431 09520 09473 09493 09491 09492 00398 00405 00398 00409 00399 00405020 09479 09581 09534 09496 09491 09491 00819 00837 00821 00817 00797 00808033 09581 09695 09669 09506 09510 09502 01437 01490 01457 01349 01316 01334050 09686 09853 09834 09518 09512 09514 02420 02615 02538 02044 01994 02022067 09776 09940 09927 09507 09506 09510 03652 04272 04089 02740 02673 02710

100

005 09454 09502 09463 09496 09496 09492 00139 00140 00138 00141 00140 00141010 09479 09528 09494 09511 09502 09507 00281 00283 00279 00283 00280 00282020 09545 09590 09554 09500 09501 09501 00578 00584 00576 00566 00559 00563033 09621 09697 09675 09493 09489 09491 01013 01034 01018 00934 00923 00929050 09758 09844 09834 09489 09486 09488 01705 01789 01757 01416 01399 01408067 09849 09946 09939 09495 09489 09492 02570 02840 02775 01896 01873 01886

Thus the 100(1minus120572) confidence interval for 1205810based on the

pivotal quantity 119876 is

1205810radic119899

radic119887le 1205810le1205810radic119899

radic119886 (27)

where 119886 119887 gt 0 119886 lt 119887 and the length of confidence intervalfor 1205810is defined as

119871 = 1205810radic119899(

1

radic119886minus

1

radic119887) (28)

In order to find the shortest-length confidence interval for 1205810

the following problem has to be solved

goal min119886119887

1205810radic119899(

1

radic119886minus

1

radic119887)

constraint int119887

119886

119891119876(119902) 119889119902 = 1 minus 120572

(29)

where 119891119876is the probability density function of central chi-

square distribution with 119899 degrees of freedom From Casella

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 9

Table 4The 95 confidence intervals for the coefficient of variationof the weight of one-month old infants

Methods Confidence intervals LengthsLower limit Upper limit

Miller 01131 01635 00504McKay 01163 01675 00512Vangel 01162 01674 00511Normal approx 01179 01689 00510Shortest 01159 01659 00500Equal-tailed 01175 01681 00506

and Berger [33 pages 443-444] the 100(1 minus 120572) shortest-length confidence interval for 120581

0based on the pivotal quantity

119876 is determined by the value of 119886 and 119887 satisfying

11988632119891119876 (119886) = 119887

32119891119876 (119887) int

119887

119886

119891119876(119902) 119889119902 = 1 minus 120572 (30)

Table 1 is constructed for the numerical solutions of theseequations by using the R statistical software [34ndash36]

23 Equal-Tailed Confidence Interval The 100(1minus120572) equal-tailed confidence interval for 120581

0based on the pivotal quantity

119876 is

1205810radic119899

radic12059421198991minus1205722

le 1205810le1205810radic119899

radic12059421198991205722

(31)

where 12059421198991205722

and 12059421198991minus1205722

are the 100(1205722) and 100(1 minus 1205722)

percentiles of the central chi-square distribution with 119899

degrees of freedom respectively

3 Simulation Study

AMonte Carlo simulationwas conducted using the R statisti-cal software [34ndash36] version 301 to investigate the estimatedcoverage probabilities and expected lengths of three proposedconfidence intervals and to compare them to the existingconfidence intervals The estimated coverage probability andthe expected length (based on119872 replicates) are given by

1 minus 120572 = (119871 le 120581 le 119880)

119872

Length =sum119872

119895=1(119880119895minus 119871119895)

119872

(32)

where (119871 le 120581 le 119880) denotes the number of simulationruns for which the population coefficient of variation 120581 lieswithin the confidence intervalThe data were generated froma normal distributionwith a known populationmean 120583

0= 10

and 1205810= 005 010 020 033 050 and 067 and sample sizes

(119899) of 5 10 15 25 50 and 100 The number of simulationruns (119872) is equal to 50000 and the nominal confidence levels1 minus 120572 are fixed at 090 and 095 Three existing confidenceintervals are considered namely Millerrsquos [7] McKayrsquos [12]and Vangelrsquos [18]

Miller

1205810isin (1205810minus 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120)

1205810+ 1199111minus1205722

radic 12058120

119899 minus 1(1

2+ 12058120))

(33)

McKay

1205810isin (1205810[(

12059421198991minus1205722

119899minus 1)120581

2

0+12059421198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(34)

Vangel

1205810isin (1205810[(

1205942

1198991minus1205722+ 2

119899minus 1)120581

2

0+1205942

1198991minus1205722

119899 minus 1]

minus12

1205810[(

12059421198991205722

+ 2

119899minus 1)120581

2

0+12059421198991205722

119899 minus 1]

minus12

)

(35)

The upper McKayrsquos limit will have to be set to infin under thefollowing condition [25]

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

) (36)

and the upper Vangelrsquos limit will have to be set to infin underthe following condition

1205810ge radic119899radic

12059421198991205722

(119899 minus 1) (119899 minus 12059421198991205722

minus 2) (37)

As can be seen from Tables 2 and 3 the three proposedconfidence intervals have estimated coverage probabilitiesclose to the nominal confidence level in all cases On the otherhand theMillerrsquos McKayrsquos and Vangelrsquos confidence intervalsprovide estimated coverage probabilities much different fromthe nominal confidence level especially when the populationcoefficient of variation 120581

0is large In other words the esti-

mated coverage probabilities of existing confidence intervalstend to be too high Additionally the estimated coverageprobabilities of existing confidence intervals increase as thevalues of 120581

0get larger (ie for 95 McKayrsquos confidence

interval 119899 = 10 09522 for 1205810= 005 09539 for 120581

0=

010 09856 for 1205810= 067) However Figure 1 shows that

the estimated coverage probabilities of the three proposedconfidence intervals do not increase or decrease according tothe values of 120581

0

As can be seen from Figure 2 McKayrsquos and Vangelrsquos con-fidence intervals have longer expected lengths than Millerrsquos

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Probability and Statistics

Shapiro-Wilk normality test

data weight

W = 0978 P-value = 03383

Algorithm 1 Shapiro-Wilk test for normality of the weights of 61one-month old infants

and the proposed confidence intervals While the expectedlengths of the three proposed confidence intervals are shorter

than the lengths of the existing ones in almost all casesAdditionally when the sample sizes increase the lengthsbecome shorter (ie for 95 shortest-length confidenceinterval 120581

0= 020 01553 for 119899 = 10 00949 for 119899 = 25 00665

for 119899 = 50)

4 An Empirical Application

To illustrate the application of the confidence intervalsproposed in the previous section we used the weights (ingrams) of 61 one-month old infants listed as follows

4960 5130 4260 5160 4050 5240 4350 4360 3930 4410 4610

4550 4460 2940 4160 4110 4410 4800 5130 3670 4550 4290

4950 5210 3210 4030 3580 4360 4360 3920 4050 4630 3756

4586 5336 2828 4172 4256 4594 4866 4784 4520 5238 4320

5330 3836 5916 5010 4344 3496 4148 4044 5192 4368 4180

4102 5210 4382 5070 5044 3530

(38)

The data are taken from the study by Ziegler et al [37] (citedin Ledolter and Hogg [38] page 287) The histogram densityplot Box-and-Whisker plot and normal quantile-quantileplot are displayed in Figure 3 Algorithm 1 shows the resultof the Shapiro-Wilk normality test

As they appear in Figure 3 and Algorithm 1 we find thatthe data are in excellent agreement with a normal distri-bution From past research we assume that the populationmean of the weight of one-month old infants is about 4400grams An unbiased estimator of the coefficient of variationis 1205810≃ 09091 The 95 of proposed and existing confidence

intervals for the coefficient of variation are calculated andreported in Table 4 This result confirms that the threeconfidence intervals proposed in this paper are more efficientthan the existing confidence intervals in terms of length ofinterval

5 Conclusions

The coefficient of variation is the ratio of standard deviationto the mean and provides a widely used unit-free measureof dispersion It can be useful for comparing the variabilitybetween groups of observations Three confidence intervalsfor the coefficient of variation in a normal distribution with aknown population mean have been developedThe proposedconfidence intervals are compared with Millerrsquos McKayrsquosand Vangelrsquos confidence intervals through a Monte Carlosimulation study Normal approximation shortest-lengthand equal-tailed confidence intervals are better than theexisting confidence intervals in terms of the expected lengthand the closeness of the estimated coverage probability to thenominal confidence level

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Professor Dr Tonghui WangProfessor Dr John J Borkowski and anonymous refereesfor their valuable comments and suggestions which havesignificantly enhanced the quality and presentation of thispaper

References

[1] K Kelley ldquoSample size planning for the coefficient of variationfrom the accuracy in parameter estimation approachrdquo BehaviorResearch Methods vol 39 no 4 pp 755ndash766 2007

[2] K Ahn ldquoOn the use of coefficient of variation for uncertaintyanalysis in fault tree analysisrdquoReliability Engineering and SystemSafety vol 47 no 3 pp 229ndash230 1995

[3] J Gong and Y Li ldquoRelationship between the EstimatedWeibullModulus and the Coefficient of Variation of the MeasuredStrength forCeramicsrdquo Journal of theAmericanCeramic Societyvol 82 no 2 pp 449ndash452 1999

[4] D S Faber and H Korn ldquoApplicability of the coefficient ofvariation method for analyzing synaptic plasticityrdquo BiophysicalJournal vol 60 no 5 pp 1288ndash1294 1991

[5] A J Hammer J R Strachan M M Black C Ibbotson andR A Elson ldquoA new method of comparative bone strengthmeasurementrdquo Journal of Medical Engineering and Technologyvol 19 no 1 pp 1ndash5 1995

[6] J Billings L Zeitel J Lukomnik T S Carey A E Blank andL Newman ldquoImpact of socioeconomic status on hospital use inNew York Cityrdquo Health Affairs vol 12 no 1 pp 162ndash173 1993

[7] E G Miller and M J Karson ldquoTesting the equality of twocoefficients of variationrdquo in American Statistical AssociationProceedings of the Business and Economics Section Part I pp278ndash283 1977

[8] D B Pyne C B Trewin and W G Hopkins ldquoProgression andvariability of competitive performance of Olympic swimmersrdquoJournal of Sports Sciences vol 22 no 7 pp 613ndash620 2004

[9] M Smithson ldquoCorrect confidence intervals for various regres-sion effect sizes and parameters the importance of noncentral

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 11

distributions in computing intervalsrdquo Educational and Psycho-logical Measurement vol 61 no 4 pp 605ndash632 2001

[10] BThompson ldquoWhat future quantitative social science researchcould look like confidence intervals for effect sizesrdquo Educa-tional Researcher vol 31 no 3 pp 25ndash32 2002

[11] J H Steiger ldquoBeyond the F test effect size confidence intervalsand tests of close fit in the analysis of variance and contrastanalysisrdquo Psychological Methods vol 9 no 2 pp 164ndash182 2004

[12] A TMcKay ldquoDistribution of the coefficient of variation and theextended t distributionrdquo Journal of the Royal Statistics Societyvol 95 no 4 pp 695ndash698 1932

[13] E C Fieller ldquoA numerical test of the adequacy of AT McKayrsquosapproximationrdquo Journal of the Royal Statistical Society vol 95no 4 pp 699ndash702 1932

[14] B Iglewicz Some properties of the coefficient of variation [PhDthesis] Virginia Polytechnic Institute Blacksburg Va USA1967

[15] B Iglewicz and R H Myers ldquoComparisons of approximationsto the percentage points of the sample coefficient of variationrdquoTechnometrics vol 12 no 1 pp 166ndash169 1970

[16] E S Pearson ldquoComparison of ATMcKayrsquos approximationwithexperimental sampling resultsrdquo Journal of the Royal StatisticsSociety vol 95 no 4 pp 703ndash704 1932

[17] G J Umphrey ldquoA comment on McKayrsquos approximation forthe coefficient of variationrdquo Communications in Statistics-Simulation and Computation vol 12 no 5 pp 629ndash635 1983

[18] M G Vangel ldquoConfidence intervals for a normal coefficient ofvariationrdquo American Statistician vol 50 no 1 pp 21ndash26 1996

[19] W Panichkitkosolkul ldquoImproved confidence intervals for acoefficient of variation of a normal distributionrdquo ThailandStatistician vol 7 no 2 pp 193ndash199 2009

[20] K K Sharma and H Krishna ldquoAsymptotic sampling distri-bution of inverse coefficient-of-variation and its applicationsrdquoIEEE Transactions on Reliability vol 43 no 4 pp 630ndash6331994

[21] E G Miller ldquoAsymptotic test statistics for coefficient of varia-tionrdquoCommunications in Statistics-Theory andMethods vol 20no 10 pp 3351ndash3363 1991

[22] K C Ng ldquoPerformance of three methods of intervalestimation of the coefficient of variationrdquo InterStat 2006httpinterstatstatjournalsnetYEAR2006articles0609002pdf

[23] R Mahmoudvand and H Hassani ldquoTwo new confidence inter-vals for the coefficient of variation in a normal distributionrdquoJournal of Applied Statistics vol 36 no 4 pp 429ndash442 2009

[24] L H Koopmans D B Owen and J I Rosenblatt ldquoConfidenceintervals for the coefficient of variation for the normal andlognormal distributionsrdquo Biometrika vol 51 no 1-2 pp 25ndash321964

[25] S Verrill ldquoConfidence bounds for normal and log-normaldistribution coefficient of variationrdquo Research Paper EPL-RP-609U SDepartment ofAgricultureMadisonWisUSA 2003

[26] N Buntao and S Niwitpong ldquoConfidence intervals for thedifference of coefficients of variation for lognormal distribu-tions and delta-lognormal distributionsrdquo Applied MathematicalSciences vol 6 no 134 pp 6691ndash6704 2012

[27] J D Curto and J C Pinto ldquoThe coefficient of variation asymp-totic distribution in the case of non-iid random variablesrdquoJournal of Applied Statistics vol 36 no 1 pp 21ndash32 2009

[28] M Gulhar B M G Kibria A N Albatineh and N U AhmedldquoA comparison of some confidence intervals for estimating the

population coefficient of variation a simulation studyrdquo SORTvol 36 no 1 pp 45ndash68 2012

[29] J A Rice Mathematical Statistics and Data Analysis DuxburyPress Belmont Calif USA 2006

[30] S F ArnoldMathematical Statistics Prentice-Hall New JerseyNJ USA 1990

[31] E J Dudewicz and S N Mishra Modern Mathematical Statis-tics John Wiley amp Sons Singapore 1988

[32] R L GrahamD E Knuth andO PatashinkAnswer to Problem960 in Concrete Mathematics A Foundation for ComputerScience Addison-Wesley Reading Pa USA 1994

[33] G Casella and R L Berger Statistical Inference Duxbury PressCalifornia Calif USA 2001

[34] R Ihaka and R Gentleman ldquoR a language for data analysis andgraphicsrdquo Journal of Computational andGraphical Statistics vol5 no 3 pp 299ndash314 1996

[35] R Development Core TeamAn Introduction to R R Foundationfor Statistical Computing Vienna Austria 2013

[36] RDevelopment Core Team R A Language and Environment forStatistical Computing R Foundation for Statistical ComputingVienna Austria 2013

[37] E Ziegler S E Nelson and J M Jeter Early Iron Supplemen-tation of Breastfed Infants Department of Pediatrics Universityof Iowa Iowa City Iowa USA 2007

[38] J Ledolter and R V Hogg Applied Statistics for Engineers andPhysical Scientists Pearson New Jersey NJ USA 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of