National College of Business Administration & Economics...
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National College of Business Administration & Economics Lahore EXPONENTIATED MOMENT EXPONENTIAL DISTRIBUTIONS BY SYED ANWER HASNAIN DOCTOR OF PHILOSOPHY IN STATISTICS MARCH, 2013
Transcript of National College of Business Administration & Economics...
National College of BusinessAdministration & Economics
Lahore
EXPONENTIATED MOMENTEXPONENTIAL DISTRIBUTIONS
BY
SYED ANWER HASNAIN
DOCTOR OF PHILOSOPHYIN
STATISTICS
MARCH, 2013
ii
NATIONAL COLLEGE OF BUSINESSADMINISTRATION & ECONOMICS
EXPONENTIATED MOMENTEXPONENTIAL DISTRIBUTIONS
BY
SYED ANWER HASNAIN
A dissertation submitted toFaculty of Social Sciences
In partial fulfillment of therequirements for the Degree of
DOCTOR OF PHILOSOPHYIN
APPLIED STATISTICS
March, 2013
iii
IN THE NAME OF ALLAHMOST BENEFICENT
AND MERCIFUL
iv
NATIONAL COLLEGE OF BUSINESSADMINISTRATION & ECONOMICS
LAHORE
EXPONENTIATED MOMENTEXPONENTIAL DISTRIBUTIONS
BYSYED ANWER HASNAIN
A dissertation submitted to the Faculty of Social Sciences, in partialfulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHYIN STATISTICS
Dissertation Committee:
_____________________Chairman
_____________________Member
_____________________Member
___________________Rector
National College of BusinessAdministration & Economics
v
DECLARATION
I, Syed Anwer Hasnain, on solemn affirmation declare that this
research work has not been submitted for obtaining a similar degree from any
other university / college.
SYED ANWER HASNAINMarch 29, 2013
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DEDICATED TO
My Late Parents,
whose desire for
my higher education
has been a beacon
light all along.
May ALLAH ALMIGHTY
bless their souls
and grant them eternalpeace in heavens!
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ACKNOWLEDGEMENTS
It is all by the grace of Allah Almighty that I could write this researchthesis. I am deeply indebted to my supervisor, Dr. Munir Ahmad, for hisbenevolent support and guidance. He led me through this arduous researchwork most patiently and was the real driving force behind it. His help withdifferent problems of analysis and description was invaluable and, hasresulted in a large number of important corrections and improvements.
It is a great pleasure to pay my special thanks to Dr. Muhammad Haniffor giving me the moral support and encouragement for the completion of thisresearch project. My special thanks are also due to Mr. Zaheer Hussain whohelped me in preparing this thesis. This thesis would be far from finishedwithout his unfailing courtesy.
I am highly grateful to Dr. Zafar Iqbal, Dr. Abdul Razzaq, Altaf urRehman and my other friends for their sincere cooperation. I thank all whohave supported me in this study in any manner or measure.
Thanks are also due to Mr. Muhammad Iftikhar for his guidance,and Muhammad Imtiaz for composing thesis.
Finally, I thanks all members of my family for their support andtolerance during this study.
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RESEARCH COMPLETION CERTIFICATE
Certified that the research work contained in this thesis entitled
‘Exponentiated Moment Exponential Distributions’ has been carried out
and completed by Syed Anwer Hasnain under my supervision during his
Ph.D. Applied Statistics Programme.
(Dr. Munir Ahmad)Supervisor and Rector,
National College of BusinessAdministration and Economics
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SUMMARY
Moment distributions have a vital role in mathematics and statistics,especially in probability theory in the perspective of research which is relatedto ecology, reliability, biomedicine and several other fields. Rao (1965)extended the basic idea of Fisher (1934) and introduced moment or weighteddistributions to model for such situations. The moment distribution deals inthe perspective of unequal probability sampling. Dara and Ahmad (2012)proposed moment exponential distribution and discussed its variousproperties.
In this research work, we propose an exponentiated momentexponential distribution (EMED) which is more flexible than Dara andAhmad (2012) moment exponential distribution and Gupta and Kundu (2001)exponentiated exponential distribution in data fitting. The two-parameterexponentiated moment exponential distribution is more effectively used inanalyzing several lifetime data than the two parameters gamma, twoparameters Weibull distribution or two parameters exponentiated exponentialdistributions. Different choices of combinations of parameters make EMEDmore useful in curve fitting. EMED produces many special cases.
The main objective of this research is to develop theoretical propertiesas well as some numerical study of EMED. We focus on some basicproperties and discuss unbaisedness and sufficiency of EMED.Characterization of EMED is presented through conditional expectation. Theproposed extension is utilized to model a data set. We also discuss differentproperties of EMED of nth order statistics and finally, we find dualgeneralized order statistics for EMED.
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TABLE OF CONTENTS
Page
DECLARATION vDEDICATION viACKNOWLEDGEMENTS viiRESEARCH COMPLETION CERTIFICATE viiiSUMMARY ix
Chapter-1: INTRODUCTION1.1 Introduction 1
Chapter-2: LITERATURE REVIEW2.1 Introduction 42.2 Order Statistics 08
Chapter-3: EXPONENTIATED MOMENT EXPONENTIALDISTRIBUTIONS (EMED)
3.1 Introduction 93.1.1 Graph of cdf of EMED
3.2 Moment Generating Function 113.3 The Factorial Moments 163.4 The Median of EMED 183.5 Survival function 19
3.5.1 Graph of Survival Function 19
3.6 Hazard Rate Function 203.6.1 Graph of Hazard Rate 20
3.7 Entropy 213.8 Information Function 223.9 Characterization 223.10 Estimation of parameters of EMED 25
Chapter-4: ORDER STATISTICS OF EMED4.1 Introduction 274.2 EMED order statistics 27
4.2.1 Graph of pdf of EMED order statistics 284.3 The Median 294.4 The Mode 294.5 Maximum Likelihood Estimation 304.6 Distribution of T and T ′ 314.7 MGF of α̂ 334.8 MGF of Dual Generalized Order Statistics of EMED 38
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4.9 Hazard Rate Function 394.9.1 Hazard Rate Graph 40
4.10 Survival Function of EMED 414.10.1 Graph of Survival Function of EMED 41
REFERENCES 42
1
CHAPTER 1
INTRODUCTION
1.1 Introduction
The exponentiated exponential distribution is a specific family of theexponentiated Weibull distribution. In analyzing several life time datasituations, it has been observed that the dual parameter exponentiatedexponential distribution can be more effectively used as compared to both dualparameters of gamma or Weibull distribution. When we consider the shapeparameter is one, then the three above mentioned distributions become thesame as one parameter exponential distribution. Hence, these threedistributions are the off shoots of the exponential distribution.
If ( )XF x is a cumulative distribution function, then ( )XF xα is
called the exponentiated distribution (ED) function, where α is the
exponentiated parameter. If X is a non-negative random variable with pdf
f(x), then the weighted distribution is defined as( )
( ), ( )
( ),
w x f xg x
E w X
β=
β (see
Patil, 2000). If ( ) mw X X= , then g(x) is called the moment distribution of X
, also named as size-biased distribution of order m . Let the random variable
X be pdf ( )f x , then the corresponding moment distribution has pdf
( ) ( ) ( )( )w
w x f xg x
E w X=
.
Dara and Ahmad (2012) proposed a cumulative distribution function ofmoment exponential distribution as
( ) ( )1 exp( ), x > 0 and > 0.
xG x x
+ β= − − β β
βIt is now natural to develop a new family of distributions named asexponentiated moment exponential distribution (EMED) or exponentiatedweighted exponential distribution (EWED) of the form
2
( ) ( )1 exp( ) , x > 0, and > 0 (1.1)X
xF x x
α+ β = − − β α β β
.
Gompertz (1825) used the cumulative distribution function to comparethe known human mortality tables and represented population growth modelwith the cdf
( ) ( )1 , , t, and > 0 (1.2)tF t eα− λ= − ρ ρ λ α .
Mudholkar et al. (1995) employed exponentiated weibull distributionto study the bus motor failure. Gupta et al. (1998) modified (1.2) bysubstituting 1ρ = . In this regard gamma, Pareto and Weibull, which are thebase line distributions, were used by them. They studied the monotonicity ofthe failure rates. Primarily, various hazard rate properties were discussedunder different situations.
Gupta and Kundu (1999) studied the theoretical properties of the threeparameters of (1.2) and generalized the exponential distribution and alsocompared them with the characteristics of gamma and the Weibulldistribution. They derived several moments and discussed maximumlikelihood estimators and a few hypotheses are tested. Finally through a dataset, it was observed that the three-parameter of generalized exponentialdistribution stands better than the three-parameter Weibull distribution or thethree-parameter gamma distribution.
Gupta and Kundu (2000) developed five estimation procedures andcompared their performances by considering two parameters generalizedexponential distribution. They developed stochastic ordering results anddiscussed the maximum likelihood estimators and their asymptotic properties.Some numerical experimental results were illustrated with the help of twodata sets.
For ( )P X Y< , where X and Y are two independent generalizedexponential random variables under the same scale parameter but differentshape parameters, Gupta and Kundu (2005) achieved the maximumlikelihood estimator and its asymptotic distribution. To obtain the asymptoticconfidence interval of [ ]P X Y< , the asymptotic distribution was used.
Shawky and Bakoban (2008) studied the lower record values ofexponentiated gamma distribution, and derived explicit expressions for the
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single, product, triple and quadruple moments. They (2008) also establishedrecurrence relations for the single, product, triple and quadruple moments andmoment generating function. Shawky and Bakoban (2009) also defined thesingle and double moments of order statistics in more comprehensive way.Based on the moments of order statistics, the best linear unbiased estimators(BLUE) for the location and scale parameters of EG distribution under Type-II censoring were obtained. The variances and co-variances of theseestimators were also presented. The maximum likelihood estimator (MLE)for the shape parameter and estimators based on order statistics for the shapeparameter were derived.
Raja and Mir (2011) worked on extension of exponentiated Weibull,exponentiated exponential, exponentiated lognormal and exponentiatedgumble distributions.
Dara and Ahmad (2012) proposed a distribution function of momentexponential distribution and developed some basic properties like moments,skewness, kurtosis, moment generating function and hazard function.
Some of work have been done on generalized weibull, exponentiatedweibull, generalized exponential, exponentiated exponential, etc. [see Guptaand Kundu (2001, 2004); Eissa (2003); Choudhury (2005); Nadraja and Kotz(2006); Carrasco et al. (2008); Barreto-Souza and Cribari-Neto (2009);Santos et al. (2010); Cordeiro et al. (2011) and Nandaraja (2011)].
This study introduces new distributions named as exponentiatedmoment exponential distribution (EMED). The probability density function(pdf) corresponding to (1.1) is
( )
1
20, 1, 0
= 0,
( ) 1 1 exp( ) exp( ),
else where. 31.
xf x x x x x
α− α= − + − β − β ββ>
α > β >
Here α is the shape parameter and β is the scale parameter. Theexponentiated moment exponential distribution (1.3) is denoted asEMED ( ),α β . In literature, no work has far been done on this area.
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CHAPTER 2
LITERATURE REVIEW
2.1 INTRODUCTION
In this chapter we present the literature review related to exponentiatedexponential distribution or generalized exponential distribution.
2.2 REVIEW OF LITERATURE
By using Gompertz’s distribution function ( ) ( )1 ,tF t eα− λ= − ρ The
exponentiated exponential distribution is derived by taking 1ρ = . Gupta andKundu (2001) defined two parameter exponentiated exponential distribution
Earlier Gupta et al. (1998) gave the concept of exponentiateddistribution by using base line distributions such as gamma, Pareto and
ordering, failure rate ordering mean residual life ordering, dispersive orderingand order statistics ordering.
Gupta and Kundu (1999) introduced three parameter generalizedexponential distribution. They calculated the hazard function and compared itwith gamma and Weibull distributions.
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By considering dual parameter generalized exponential distribution,Gupta and Kundu (2000) compared the MLE's with the other estimators such
Gupta and Kundu (2001) considered a two-parameter exponentiated
They (2001) proposed that instead of Weibull and gamma distribution,generalized exponential distribution is equally applicable.
By using two different methods, Gupta and Kundu (2003a) checked thecloseness between the gamma distribution and the generalized exponential
distinguishable in case of α close to one.
Gupta and Kundu (2003b) used the proportion of the maximized
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Gupta and Kundu (2004) also used the ratio of the maximized
used these asymptotic results to determine the minimum sample size.
By having the same scale parameter, Gupta and Kundu (2005) stated
scale parameter given or not given. If it is not given, then maximumlikelihood estimator works effectively. Construction of confidence interval is
From an exponentiated gamma distribution, Shawky and Bakoban(2008c) discussed lower record values. They (2008c) derived explicitexpressions for the single, product, triple and quadruple moments. The singlemoments of the thn lower record value ( )XL n from exponentiated gammadistribution is
Similarly they showed triple and quadruple moments.
They (2008c) also established recurrence relations for the single moment as
7
( ) ( ) ( )21 1
3
11 ,
2 !a a ia
n n ni
a a
i
∞ + −+ +
=
µ − + µ = µ θ θ ∑
Recurrence relations for the bivariate moments as
( ) ( ) ( ), 2,,, 1, 1, 1
2
1, m,n>0,
!a b a i ba b
m n m n m ni
a
i
∞ + −+ + +
=µ − µ = µ
θ∑
Similarly they obtained the recurrence relations for the triple andquadruple moments.
Raja and Mir (2011) conducted the numerical study by taking the eightdistributions namely gamma, Weibull, lognormal, gumble, exponentiatedWeibull, exponentiated exponential, exponentiated lognormal andexponentiated gumble distribution have been fitted to two real life data sets.
The first data set is regarding the failure times of the conditioningsystem of an aeroplane. Exponentiated exponential and gamma give betterresults followed by Weibull. Hence gamma, Weibull and exponentiatedexponential can be replaced as alternative to each other in certain situations.In second data set regarding runs scored by a cricketer, under exponentiatedlognormal and exponentiated exponential (Raja & Mir, 2011) give better fitfollowed by gamma. Hence exponentiated lognormal, exponentiatedexponential and gamma can be replaced as an alternative with each other.Since the distribution function of exponentiated lognormal distribution can beexpressed in terms of normal distribution, therefore, inference based on thedata can be handled easily and can be given preference.
Dara and Ahmad (2012) defined the cumulative distribution function(cdf) of size-biased moment exponential distribution as:
( )( ) ( )exp
1xx
G xβ+ β −
= −β
.
and developed thr moment about the origin( )( )2
2
r
rrβ Γ +′µ =
Γ. They (2001)
also derived the characteristic function, ( ) ( ) 22 1MX t it
−−βϕ = β − and the
hazard rate function as ( ) ( )gx
h xx
=β + β
.
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2.3 ORDER STATISTICS
Related to life testing studies, order statistics definitely exists inmost of the real-life problems involving data. In the study of order statistics,David (1981), Balakrishnan and Cohen (1991), Arnold et al. (1992), andDavid and Nagaraja (2003) have drawn inference. On the basis of orderstatistics, Balakrishnan and Puthenpura (1986) derived the best linearunbiased estimators (BLUE’s) of the location and scale parameters of halflogistic distribution. Maximum likelihood estimators (MLE’s) of theparameters of Burr XII distribution were derived for Type-II censored sampleby Wingo (1993). On the basis of order statistics, Balakrishnan and Chan(1998) have observed linear estimation for log-gamma parameters. On thebasis of order statistics with regard to Burr X distribution, Raqab (1998) hasobtained two types of estimators of location and scale parameters. Gupta and
parameters logarithmically decreasing survival distributions under Type-IIcensored samples were determined by Sultan et al. (2002). Also, Mahmoud etal. (2003) have derived the exact expression for the single and double
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CHAPTER 3
EXPONENTIATED MOMENTEXPONENTIAL DISTRIBUTIONS
3.1 INTRODUCTION
This chapter introduces a family of distributions, called exponentiatedmoment distribution (EMED). We develop its basic but standard propertiesincluding survival function and hazard rate function. Numerical study ofsome of the fundamental properties is also conducted. Characterization ofEMED through conditional moment is done.
The graphs of cumulative distribution function are plotted by usingsoftware Mathematica, when 1,2,3α = and 1,2,3β =
10
11
3.2 MOMENT GENERATING FUNCTION
After substitution we obtain
By using the binomial expansion, it reduces to
( ) ( )( )
1
20 0
1 21
i 1 ( )
ii
ji j
i j
j i t
α−
+= =
α − Γ + = α − + − β
∑ ∑ . (3.3)
12
Suppose X denotes the EM random variable with parametersα and β , then the rth raw moment of X is the coefficient of !rt r ,
( )1
12
0
1 1r x r xxE X e x e dx
α−∞− β + − β α = − + ββ
∫ ,
After substitution, we obtain
By using the binomial expansion, after simplification it reduces to
(3.4)
First moment about zero
Second moment about zero
( ) ( ) ( )( )
12 2
40 0
1 41
i 1
ii
ji j
i jE X
j i
α−
+= =
α − Γ + = αβ − +
∑ ∑ .
Third moment about zero
( ) ( )( )
13 3
50 0
1 5( ) 1
i 1
ii
ji j
i jE X
j i
α−
+= =
α − Γ + = αβ − +
∑ ∑ .
13
Fourth moment about zero
( ) ( )( )
14 4
60 0
1 6( ) 1 .
i 1
ii
ji j
i jE X
j i
α−
+= =
α − Γ + = αβ − +
∑ ∑
Numerical values of variances are being calculated by usingMathematica at different values of α and β .
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15
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3.3 The Factorial MomentsThe factorial moments of X are given as follows
Let X be an exponentiated moment exponential (EME) randomvariable with parameters α and β , then its negative rth moment can bewritten as
,
Let the random variable X follow the EMED with pdf
17
Differentiate it w.r.t. x
The mode of EMED for different values of α and β are given in table 3 byusing R.
The graphs of mode are plotted by using software Mathematica, when 1α =and 1,2,3 and 4β =
18
3.4 The Median of EMED
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3.5 Survival Function
3.5.1 Graph of Survival FunctionThe graph of survival function is drawn for different values ofα and β .
Fig. 3: The graph of survival functionThe survival curves show the decreasing rate.
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3.6 Hazard Rate FunctionThe hazard rate function is a significant quantitative exploration of life
phenomenon. In the literature Barlow et al. (1963) introduced hazard ratefunction in studying the relationship between properties of a distributionfunction (or density function) and corresponding hazard rate function. It hassignificant value in the study of reliability analysis, survival analysis,actuarial sciences and demography, in extreme value theory and in durationanalysis in economics and sociology.
For the EMED, the hazard rate becomes
3.6.1 Graph of Hazard Rate
Fig. 3:The graph of hazard (failure) rate functionfor selected values of α and β .
The above graph of different shapes of hazard function shows that it isan increasing function of x . Furthermore as 0x → , ( ) 0h x → and as
x → ∞ , ( )h x → ∞ .
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3.9 Characterization
An important area of statistical theory is characterizations ofprobability distributions. Different methods are used for the characterizations
Theorem 3.1
( ) ( )1 exp 0, 1, , 0x
F x x xα > α >+ β= − − β β
β >
If and only if
Proof
The necessary part follows as
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By performing integration by parts, we obtain
( ) ( ) ( ) ( )1 1 exp1 exp
1
t tt t
− + β − β = + β − β +α +
( ) ( )1 1 exp
1
t t+ α + β − β =α +
For sufficient case
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after simplification
Integrating the above function
which proves that
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3.10 Estimation of the parameters of EMED
( )1, ; ,... ln 2 ln ln ( 1) ln, 1 1 jxjjn
jx xn n xln x x eL
− βΣ = α − β + − + α − Σ −α β + β β
(3.7)
(3.8)
Equating (3.7) to zero, we have the following MLE α becomes
( )ˆ
ln 1 1 x
n
x e− β−α =
Σ − + β .
Again differentiate (3.8) w.r.t α
2
2 2
ln L n∂ −=∂α α
Hence variance is as follows
2
( )Vn
αα =
Now differentiate equation (3.4) w.r.t. Beta, we obtain
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(3.8)
By solving (3.8) the MLE for β is obtained.
Again differentiating equation (3.8) w.r.t. Beta
From the above equation variance of beta is obtained.
Now differentiateln L∂∂α
w.r.t. Beta
.
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CHAPTER 4
ORDER STATISTICS OF EMED
4.1 INTRODUCTION
There are many problems of statistical inference in which one is unableto assume the functional form of the population distribution. Many of theseproblems are such that the strongest assumption which can be reasonablymade is continuity of the cumulative distribution function of the population.An increasing amount of attention is being devoted to statistical tests whichhold for all populations having continuous cumulative distribution functions.In such problems it is being found that order statistics, that is, the ordered setof values in a random sample from least to greatest, are playing afundamental role. It is to be particularly noted that the term order is not beingused here in the sense of arrangement of sample values in a sequence as theyare actually drawn.
In this chapter, order statistics of exponentiated moment distribution(EMED) are considered. We discuss some properties of distribution of orderstatistics of moment distribution and exponentiated moment distribution.Finally, we find dual generalized order statistics for EMED.
4.2 EMED ORDER STATISTICS
Let 1,..., nX X be independent and identically distributed exponentiated
n variables. The thn order statistic, or the sample maximum, ( )nX has pdf
given by
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which is pdf of EMED nth order statistic.
4.2.1 Graph of PDF of EMED Order Statistics
The following graphs are shown for different values of parameters.
Fig. 4.1: The Graph of f(z) of Ordered Function
5 10 15 20 25
0.05
0.10
0.15
0.20
0.25
0.30n20,3,3
n10,2,2
n5,1,1
f(z)
z
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4.3 MEDIAN
The median of EMED thn order statistics can be found as
(4.2)
equation (4.2) can be solved through different softwares numerically for fixedvalues of parameters.
4.4 MODE
Differentiating equation (4.1) we have
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4.5 MAXIMUM LIKELIHOOD ESTIMATION
Equating it to zero
In the next sections we find distribution of T and T ′ .
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4.6 DISTRIBUTION OF T AND T ′
The following theorems illustrate the distributions of T and T ′ .
Theorem 4.1If ~ ( , )X EMED α β is random variable with β known, then
( ) ( )ln 1 1 expT X X= − − + β − β follows exponential function.
Proof
As
Jacobian of transformation is
The probability distribution function of T is
The mean and variance of T are1
andα 2
1
αrespectively.
Theorem 4.2
If ( )ln 1 1 ixi iT x e− β = − − + β are iid random variables, then
( ) ( ) ( )1ln 1 1 exp ~ ,iT T x x G n
−′ = =Σ − + β − β α ∑ .
ProofSince T ~i exponential r.v., then by additive property of exponential
distribution ( )1
~ ,n
ii
T T G n=
′ = α∑ .
Corollary
32
Theorem 4.3
If then1T ′
is an
inverted Gamma Distribution.
Theorem 4.4
( )1
( ) 1 ln 1 xi x eT
−− β = − + β transformed ordered failures follows
Exponential function.
Proof
As 1 2, , , nX X X… are iid with EMED (α ,β ) then iT transformedordered failures time are also iid follows exp( α ).
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4.7 MOMENT GENERATING FUNCTION OF α
Since 1 2, , , nZ Z Z… are iid, then
( )| , , 0tg z e z−αα = α α ≥
then
( )ˆ
ii
n n
T Zα = =
∑ ∑
The distribution ofˆ 1
~iZ
Y Gamman α
α= =α ∑
34
The m.g.f. of iZ∑ is
It can also be written as
( ) ( )1i
nZM t t
−α = −∑
By using the uniqueness theorem iZα∑ follows Gamma (n) andˆ 1
iZ
Yn α
α= =α ∑
follows inverted Gamma Distribution.
The pdf of α̂ is given as
Moments of α̂
The thr moment of α̂ is given as
35
Clearly ( )ˆE α ≠ α , this shows that MLE of α is not an unbiasedestimate of α but it is asymptotically unbiased estimate of α .
Theorem 4.6
Proof:
By the thn order statistic, or the sample maximum, ( )nX has pdf as
Let
( ) ( )1 1 1 exp nyx x
n
α− − + β − β =
By differentiation the above expression, we obtained
( )( )1
( ) = 1n
nn
yg y
n
α − −
Its distribution function is
36
after simplification we obtained
( )( )1 1
( )
11
-1 1 1 1
1 1
n
n
y
nG y
n n n n
α − + − = α − + α − +
As n → ∞ , then
and
( )(n)g = nyy e−α
Hence proved
Theorem 4.7
Limiting distribution of ( )1n rX − + . Limiting distribution for the thr
largest order statistic ( )1n rX − + for fixed r is a Gamma Distribution under a
special transformation.
37
Proof:
More generally the thr order statistic ( )jX has pdf
Replacing r by 1n r− +
Let
Y1 1xx
en
α− β+ β − = − β
Simplified form
As n → ∞ , then
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4.8 MGF DUAL GENERALIZED ORDER STATISTICS OF EMED
Moment Generating function of dual generalized order statistics
After replacing EMED
Hence Moment Generating function of dual generalized order statisticsEMED
39
is the required mgf of dgos of EMED.
4.9 HAZARD RATE FUNCTION
Hazard Rate Function arises in the situation of the analysis of the timeto the event and it describes the current chance of failure for the populationthat has not yet failed. This function plays a pivotal role in reliabilityanalysis, survival analysis, actuarial sciences and demography, in extremevalue theory and in duration analysis in Economics and Sociology. This isvery important for researchers and practitioners working in areas likeengineering statistics and biomedical sciences. Hazard rate function is veryuseful in defining and formulating a model when dealing with lifetime data.
.
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4.9.1 Hazard Rate Graph
Fig. 4.2: The Graph of ( )h x of Hazard Function of Ordered Statistics
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4.10 SURVIVAL FUNCTION OF EMED
The branch of Statistics that deals with the failure in mechanicalsystems is called survival analysis. In engineering it is called as reliabilityanalysis or reliability theory. In fact the survival function is the probability offailure by time y , where y represents survival time. We use the survivalfunction is to predict quantiles of the survival time. By definition of survivalfunction
4.10.1 Graph of Survival Function of EMED
Fig. 4.3: The Graph of ( )S z of Survival Function of Ordered Statistics
42
REFERENCES
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2. Balakrishnan, N. and Cohen, A.C. (1991). Order Statistics andInference: Estimation Methods, Academic Press, San Diego.
3. Balakrishnan, N. and Chan, P.S. (1998). Log-gamma order statisticsand linear estimation of parameters, In: Balakrishnan, N. and Rao, C.R.(Eds.), Handbook of Statistics, Vol. 17. Elsevier Science, Amsterdam,61-83.
4. Balakrishnan, N. and Puthenpura, S. (1986). Best linear unbiasedestimators of location and scale parameters of the half logisticdistribution. J. Statist. Comput. Simul., 25, 193-204.
5. Barlow, R.E., Marshall, A.W. and Proschan, F. (1963). Properties ofProbability Distributions with Monotone Hazard Rate. Ann. Math.Statist., 34(2), 375-389.
6. Dara, S.T. and Ahmad, M. (2012). Recent Advances in MomentDistributions and their Hazard Rate. Ph.D. Thesis. National College ofBusiness Administration and Economics, Lahore, Pakistan.
7. David, H.A. (1981). Order Statistics, Second Edition. Wiley, NewYork.
8. David, H.A. and Nagaraja, H.N. (2003). Order Statistics, ThirdEdition. Wiley, New Jersey.
9. Fisher, R.A. (1934). The Effects of Methods of Ascertainment uponthe Estimation of Frequencies. Annals of Eugenics, 6, 13-25.
10. Gompertz, B. (1825). On the nature of the function expressive of thelaw of human mortality and on a new mode of determining the value oflife contingencies. Philosophical Transactions of the Royal SocietyLondon, 115, 513-585.
11. Gupta, R.C.; Gupta, R.D. and Gupta, P.L. (1998). Modeling failuretime data by Lehman alternatives. Commun. in Statist. Theo. andMeth., 27(4), 887-904.
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