79e4150bf71d4edcd1

Communications in StatisticsSimulation and Computation, 36: 643656, 2007Copyright Taylor & Francis Group, LLCISSN: 0361-0918 print/1532-4141 onlineDOI: 10.1080/03610910701207819

Quality Control

Acceptance Sampling Plans from TruncatedLife Tests Based on the GeneralizedBirnbaumSaunders Distribution

N. BALAKRISHNAN1, VCTOR LEIVA2,AND JORGE LPEZ3

1Department of Mathematics and Statistics, McMaster University,Hamilton, Ontario, Canada2Department of Statistics, CIMFAV, Universidad de Valparaso,Valparaso, Chile3Management of Process Engineering, MOLYMET S.A. (Molybdenumand Metals), Santiago, Chile

In this article, we develop acceptance sampling plans when the life test is truncatedat a pre-xed time. The minimum sample size necessary to ensure the speciedmedian life is obtained by assuming that the lifetimes of the test units follow ageneralized BirnbaumSaunders distribution. The operating characteristic values ofthe sampling plans as well as producers risk are presented. Two examples are alsogiven to illustrate the procedure developed here, with one of them being based ona real data from software reliability.

Keywords Acceptance sampling; BirnbaumSaunders distribution; Consumersrisk; Kurtosis; Life test; Minimum sample size; Producers risk.

Mathematics Subject Classication 62N05; 90B25; 62P30; 62H10.

1. Introduction

Quality is now not only an option or aim of companies, but a necessity forbusinesses in a global market. Thus, the quality has become a differentiation toolbetween competitive enterprises. Two important tools for ensuring quality are thestatistical quality control and the acceptance sampling.

The acceptance sampling is a sampling inspection in which the consumer decidesto accept or to reject a lot of products shipped by the producer, based on the

Received July 13, 2006; Accepted December 15, 2006Address correspondence to Vctor Leiva, Department of Statistics, CIMFAV,

Universidad de Valparaso, Valparaso, Chile; E-mail: [email protected] or [email protected]

643

644 Balakrishnan et al.

results of a random sample selected from that lot. An acceptance sampling plan is aspecic plan that establishes the minimum sample size to be used and the associatedacceptance and non-acceptance criteria for the lot. So, an acceptance sampling planconsists of the number of units on test (n) and the acceptance number (c) such thatif there are at most c failures out of n, the lot is accepted. The consumers andproducers risks are then the probabilities that a bad lot is accepted and a goodlot is rejected, respectively; for more details, one may refer to Duncan (1986) andStephens (2001).

Life tests are experiments carried out in order to obtain the lifetime of anitem (i.e., time to its failure or the time it stops working satisfactorily). A commonpractice in life tests is to terminate the test at a pre-xed time, and recording thenumber of failures that occurred during that time period. In many situations, thequality of a product is measured through the lifetime (T ), and the variance orthe scale parameter of the distribution of T may serve as a quality parameter.We are especially interested in determining the probability that an unit which hassatisfactory operation during the experimental time is classied as a non defectiveunit. The acceptance sampling procedures can therefore be applied to life tests.

If the units in the lot are classied as defective or non defective based on alife testing experiment, then the acceptance sampling plan must consider a thirdelement, i.e., the ratio t

0, where t is the pre-xed test time and 0 is the specied

mean or median lifetime. Thus, a random variable (r.v.) that represents the lifetimeof the inspected unit is related to an acceptance sampling based on truncated lifetests (ASTLT).

Most of the probabilistic models used to describe lifetime data are chosen forone or more of the following reasons (see Tobias, 2004, Sec. 8.2.1): (1) a physical orstatistical theoretical argument for the mechanism of failure of the unit; (2) a modelthat has previously been used successfully; and (3) an appropriate model whoseempirical t is good to data. Moreover, Cox and Oakes (1984, p. 24) establishedthat there are also other ways to identify the suitable life distribution. For example,an analysis of the probability density function (pdf) is not always the best thing,and it may be preferable to analyze (4) the hazard function (ht) or log[ht] versust or log(t), or (5) the cumulative hazard function (Ht) or log-survival versus t orlog(t). Various families of parametric life distributions, that have some exibility, treasonably well in the central part of the distribution, but poorly in the tails. Thisis so because usually we nd little data from the tails while a reasonable amountfrom the central part, which makes it difcult for the identication of an appropriatedistribution, and so an empirical justication must be accompanied by some otherarguments. Thus, whatever justication is used to choose the life distribution, themodel must be logical and, in addition, must possess visual and statistical tests fortting the data and/or criteria for model selection.

Several products that must be put under inspection for acceptance samplingare exposed to cumulative degradation during their lifetimes. This degradationis recognized like fatigue and, in some cases, this development is intangible.Therefore, it is necessary to consider methods of acceptance sampling from lifemodels that account for this degradation and that are exible when we t thelifetime data.

An important lifetime model originating from a physical problem (materialfatigue) is the one derived by Birnbaum and Saunders (1969). The BirnbaumSaunders (BS) distribution has a close relation to the normal distribution and has

Acceptance Sampling Plans from Truncated Life Tests 645

many applications in a wide variety of contexts; see, for example, Johnson et al.(1994, p. 651). A more general derivation of this distribution was provided byDesmond (1985) based on a biological consideration. He also strengthened thephysical justication for the use of this distribution by relaxing some assumptionsmade earlier by Birnbaum and Saunders (1969). Recently, Daz-Garca andLeiva (2005, 2007) generalized the BS distribution and obtained a more exibledistribution. The main motivations for the use of the generalized BirnbaumSaunders (GBS) distribution are to make the kurtosis more exible (compared tothe BS model) and also to admit bimodality. These are achieved by means of asymmetrical distribution with different degrees of kurtosis (like the Student-t model)instead of the normal model used in the derivation of the BS distribution. Moreover,the BirnbaumSaunders distribution is a particular case of the GBS distribution.

Acceptance sampling based on truncated life tests were discussed by Epstein(1954) and Sobel and Tischendrof (1959) for the exponential model. An extensionof this work was carried out by Goode and Kao (1961) by considering the Weibullmodel which includes the exponential distribution as a particular case. Gupta andGroll (1961) and Gupta (1962) considered the gamma and log-normal distributions,respectively. More recently, Kantam et al. (2001), Baklizi (2003), Baklizi and ElMasri (2004), and Rosaiah and Kantam (2005) have all developed acceptancesampling plans for a variety of distributions. However, in these last group ofarticles, there seems to be some confusion respect to the ratio t

0, which has resulted

in certain conceptual errors that have affected their results, interpretations andexamples.

In acceptance sampling based on truncated life tests, we assume the following:(1) the units are destructible or are degraded after the life test, and (2) there areseveral distributions that model the product life reasonably well. Thus, consideringsimilar risk and operating conditions and the assumptions (1) and (2), the consumerwill benet with smaller number of units required to test. For this reason, we coulduse a distribution that gives the smallest sample size.

In this article, based on Student-t generalized BirnbaumSaunders (GBS(t))distribution, we present a methodology to nd the minimum sample size necessaryto ensure a specied median life. With a pre-xed time t, the life testing experimentwill be terminated at time t if the c + 1th failure does not occur by this time andotherwise at the c + 1th failure time. The lot is accepted if the specied medianlife can be established with a pre-xed probability (P), specied by the consumer.In Sec. 2, we describe the GBS distribution. In Sec. 3, we develop the proposedacceptance sampling plans and the operating characteristic values. The numericalresults and two illustrative examples are presented in Sec. 4, and nally, someconcluding remarks are made in Sec. 5.

2. The GBS Distribution

The BS distribution is dened in terms of the normal distribution by means ofthe random variate T = Z/2+Z/22 + 12, where Z = 1

T/ /T

N0 1, > 0 is the shape parameter, and > 0 is the scale parameter. This isdenoted by T BS . Recently, Daz-Garca and Leiva (2005, 2007) presenteda generalization of the BS distribution by assuming that Z follows a symmetricaldistribution in (elliptically contoured distributions) with location parameter = 0and scale parameter = 1, i.e., Z EC0 1 f , where f is the pdf of Z. In thiscase, the notation T GBS f will be used.


2.1. Some Characteristics of the GBS Distribution

The following results hold for the GBS distribution.If T = Z/2+Z/22 + 12 GBS f , then:

(1) Z = 1T/ /T EC0 1 f ;

(2) The pdf of T is

fT t = fat d

dtat t > 0 > 0 > 0 (1)

where f is the pdf de Z EC0 1 f ,

at =1

(t

t

)and

d

dtat =

t32 t + 2

= 12

(t

+ 1

)(t

) 32

(2)

(3) cT GBS c f , with c > 0;(4) T1 GBS 1 f ;(5) For the r.v. Z given in (1), if Zr exists, then

Tn = nn

k=0

(2n2k

) ks=0

(ks

)Z2n+sk

(

2

)2n+sk (3)

From (3), the mean and variance of T can be obtained as

T = 2

(2+Z22) and VarT = 2

4

(24Z4+ 42Z2 4Z22)

(4)

where Z EC0 1 f ;(6) The cumulative distribution function (cdf) of T is

FTt = T t = F(at

) t > 0 (5)

where F is the cdf of Z EC0 1 f and at is as given in (2);(7) The pth percentile of the GBS distribution, tp = F1T p, is given by

tp =

4

(zp +

2z2p + 4

)2 (6)

where zp is the pth percentile of Z EC0 1 f ; If p = 05, then t05 = , andso is the population median. This result is similar to that of Chang and Tang(1994) for the BS distribution.

In Table 1, the pdf and cdf of the BS and GBS(t) distributions are presented,while in Table 2, the mean and variance of the BS and GBS(t) distributions aregiven.

From Table 2, we note that if , then the mean and variance ofT GBS t converge to the mean and the variance of the classical BSdistribution, as we would expect.


Table 1pdf and cdf of T BS and T GBS f > 0 with f

being the pdf of Z EC0 1 f f pdf cdfNormal 1

2exp

[ 122 ( t + t 2)] t32 t+2

(at

)

Student-t

(+12

)

12

(2

)(1+ 12

[t+

t 2]) +12 t 32 t+

212

> 0 t(at

)Here, and t are the cdfs of the standard normal and Student-t distributions,

respectively, and at is as given in (2), where

tz =12

[1+ I z2

z2+

(1212

)]

and Ixa b is the incomplete beta function ratio (see Johnson et al., 1994, p. 364), given byIxa b =

x0 t

a11tb1dt 10 t

a11tb1dt .

2.2. The GBS Distribution as a Lifetime Model

An useful function in lifetime analysis is the failure rate or hazard function, denedby ht = ft/1 Ft, where f and F are the pdf and cdf, respectively; see,for example, Johnson et al. (1995, p. 640). The behavior of ht allows one tocharacterize the aging of the units. For example, if the failure rate is increasing (IFRdistribution), then the units age with time. If ht is decreasing (DFR distribution),then the units improve in performance with time. Finally, if ht is constant, thenthe life distribution is necessarily exponential.

The normal distribution belongs to the IFR family. However, the lognormalmodel does not have a increasing failure rate, because it is initially increasinguntil its critical point and then it decreases to zero. For the hazard function ofthe classical BS distribution, we have that: (1) behaves similarly to the lognormalmodel, but this decreases until it becomes stabilized in a positive constant (not atzero), see Nelson (1990, p. 70); (2) its behavior is similar to that of the inverseGaussian distribution, see Chhikara and Folks (1989, p. 153); (3) one cannot assumethat this is always increasing while analyzing a typical fatigue data; and (4) itsaverage is nearly nondecreasing ( IFRA distribution). For more details about (3)and (4), see Birnbaum and Saunders (1969). The class of hazard functions which are

Table 2Mean and variance of T BS and T GBS f > 0

with f being the pdf of Z EC0 1 f f T = VarT Normal

(1+ 122

)22

(1+ 542

)Student-t

(1+ 12 2 2

) > 2 22

(

2 + 54 28/5

422 2) > 4


nondecreasing on the average has been studied by Birnbaum et al. (1968), who havediscussed some closure properties that are physically plausible.

If T BS , it is easy to verify that the behavior of its hazard function,hTt, does not depend on the parameter . On the other hand, for any , we havethat: (1) hTt is unimodal, being increasing for t < tc and decreasing for t > tc,where tc is the critical point of hTt; and (2) hTt approach to 2

21 as t .In addition, (3) when 0, then hTt tends to be increasing. For more detailsabout (1)(3) see Chang and Tang (1993). Thus, it is interesting to know from whichvalue of the failure rate of T belongs to the IFR class. Although it is not possibleto obtain an analytical result in order to response to this question, a numericalstudy indicates that the BS distribution belongs to the IFR family when < 041and 0 < t < 8, which implies an IFRA class. Also, Birnbaum and Saunders (1969)showed by numerical calculation that the average failure rate of T decreases slowlyfor t < 164.

If T GBS f , then the hazard function of T is given by

hTt =(

fat

Fat )d

dtat (7)

where f and at are as given in (1), and F is as in (5). Besides,Fat = 1 Fat due to the symmetry of F. The behavior of hTtgiven in (7) is similar to that of the classical BS model for all f and in uniquemanner when the symmetric distribution in is unimodal. Expressions for hTtwhen T BS or T GBS t GBS t, with f being the pdf ofZ t, can be easily obtained from (7) and Table 1.

3. Acceptance Sampling Plans

We now assume that the lifetime (T ) of the product under study follows a GBS(t)distribution with parameters and known (xed). A common practice in lifeexperiments is to stop the life test at pre-xed time (t) and register the number offailures. One object of these tests is to set a condence (lower) limit on the mean ormedian life (or any other percentile of the distribution). It is then desired to establisha specied mean or median life with a probability of at least P (consumers risk).Normally, the mean life is used if the life distribution is symmetric, and the medianlife is used if the life distribution is skewed; see, for example, Gupta (1962). Thedecision to accept the specied mean or median life occurs if and only if the numberof failures at the end of the pre-xed time t does not exceed a given number c. Thus,the test is terminated at time t or at the c + 1th failure, whichever occurs rst.For such a truncated life test and the associated decision rule, we are interested inobtaining sampling plans, i.e., we want to nd the minimum sample size necessaryto achieve our objective.

As mentioned earlier, an acceptance sampling plan based on truncated life testsconsists of: (1) the number of units on test (n); (2) the acceptance number (c), suchthat if at most c failures out of n occur at the end of the pre-xed time t, thelot is accepted; and (3) the ratio t

0, where 0 is the specied mean or median life

and t is the maximum test duration. Thus, similar to the case of the log-normaldistribution (see Gupta, 1962), for T GBS f , 0 will represent the median,that is, 0 = 0, where is also the lot quality parameter, since is the median life


as well as the scale parameter of the GBS distribution. It is important to mentionhere that this is contrary to Baklizi (2004), who has developed ASTLT incorrectlyby mentioning and interpreting like the average (mean). Thus, an acceptancesampling plan based on truncated life tests for the GBS distribution is n c t

0.

3.1. Minimum Sample Size

We x the probability of accepting a bad lot (consumers risk), i.e., the one forwhich the true median life is below the specied 0, not to exceed 1 P. Weassume that the lot size (N ) is large enough to be considered innite (for example,n/N 010; see Stephens, 2001, p. 43), so that the binomial distribution can beused. Thus, the acceptance and non acceptance criteria for the lot are equivalent tothe decisions of accepting or rejecting the hypothesis 0. We want to nd theminimum sample size (n) such that

cx=0

(nx

)px1 pnx 1 P (8)

where p = FTt given in (5) is monotonically increasing on t and decreasing on ,for xed t, which is easy to establish for the GBS distribution. Thus, p = FTt =FTt = FT t depends only on the ratio t , if we x and . Hence, it is sufcientto specify just this ratio. Therefore, if the number of observed failures is at mostc, from (8) we can establish with probability P that FT

(t

) FT ( t0), which implies

that 0.The minimum values of n satisfying (8) for P = 075 09 095 099 and

t0

= 0628 0942 1257 1571 2356 3141 3927 4712 were determined and arepresented in Table 3, for xed and . These choices of P and t

0allow us to

compare our results with those obtained by Gupta and Groll (1961), Gupta (1962),Kantam et al. (2001), and Baklizi and El Masri (2004).

3.2. Operating Characteristic of the Sampling Plan n c t0

The operating characteristic (OC) function of the ASTLT plan n c t0 gives the

probability that the lot can be accepted. For the ASTLT plan under analysis, thisprobability is given by

Lp =c

x=0

(nx

)px1 pnx (9)

where p given in (8) is a monotonically decreasing function of 0, for xed t,while Lp is decreasing in p. Based on (9), the operating characteristic values, as afunction of

0, for xed and , were determined and are presented in Table 4 for

the ASTLT plan n c t0, for xed c and different values of P. For given P and

t0, the choice of c and n can be made on the basis of the OC function.

3.3. Producers Risk

The producers risk is dened as the probability of rejecting of the lot when 0.For the sampling plan under consideration and a given value for the producers


Table 3Minimum sample size necessary to assert that the median life exceeds a given

value, 0, with probability P and the corresponding acceptance number, c, using

binomial probabilities (when = 1 and = 5)t0

P c 0.628 0.942 1.257 1.571 2.356 3.141 3.972 4.712

0.75 0 4 3 2 2 1 1 1 11 8 5 4 4 3 3 2 22 11 8 6 5 4 4 4 33 15 10 8 7 6 5 5 54 18 12 10 9 7 6 6 65 22 15 12 10 8 8 7 76 25 17 14 12 10 9 8 87 28 19 15 13 11 10 9 98 32 22 17 15 12 11 11 109 35 24 19 17 14 12 12 1110 38 26 21 18 15 14 13 12

0.90 0 6 4 3 3 2 2 1 11 11 7 5 5 4 3 32 15 10 8 7 5 4 4 43 19 12 10 8 7 6 5 54 23 15 12 10 8 7 7 65 26 18 14 12 9 8 8 76 30 20 16 14 11 10 9 97 34 22 18 15 12 11 10 108 37 25 20 17 14 12 11 119 41 27 22 19 15 13 13 1210 44 30 24 20 16 15 14 13

0.95 0 8 5 4 4 2 2 2 21 13 8 6 5 4 4 3 32 17 11 9 7 6 5 5 43 21 14 11 9 7 6 6 54 26 17 13 11 9 8 7 75 29 19 15 13 10 9 8 86 33 22 17 15 12 10 10 97 37 25 19 17 13 12 11 108 41 27 21 18 15 13 12 119 45 30 23 20 16 14 13 1310 48 32 25 22 17 15 14 14

0.99 0 12 8 6 5 3 3 2 21 18 11 9 7 5 4 4 42 23 15 11 9 7 6 5 53 27 18 14 11 9 7 7 64 32 21 16 13 10 9 8 85 36 23 18 15 12 10 9 96 40 26 20 17 13 12 11 107 44 29 23 19 15 13 12 118 48 32 25 21 16 14 13 129 52 34 27 23 18 16 14 1410 56 37 29 25 19 17 16 15


Table 4OC values for the ASTLT plan n c t

0 for a given P when c = 2,

= 1 and = 5

0

P n t0

2 4 6 8 10 12

0.75 11 0.628 0.815159 0989668 0.998873 0999802 0999952 0.9999858 0.942 0.708606 0972787 0.996133 0999205 0999787 0.9999316 1.257 0.687864 0962127 0.993584 0998519 099957 0.9998525 1.571 0.666564 0950509 0.990339 0997543 0999235 0.9997234 2.356 0.589382 0910425 0.977004 0992896 0997449 0.9989684 3.141 0.408615 0810193 0.937440 0977013 0990622 0.9958164 3.942 0.278670 069577 0.878359 0948572 0976662 0.9886793 4.712 0.504571 0822568 0.931527 097089 0986528 0.993304

0.90 15 0.628 0.658067 0974963 0.997070 0999470 099987 0.99996110 0.942 0.562419 0948937 0.992238 0998360 0999555 0.9998548 1.257 0.477582 0914205 0.983907 0996117 0998847 0.9995987 1.571 0.400057 0871118 0.971202 0992207 0997494 0.9990745 2.356 0.385076 0826696 0.950547 0983896 0994042 0.9975455 3.141 0.408615 0810193 0.937440 0977013 0990622 0.9958164 3.942 0.278670 0695770 0.878359 0948572 0976662 0.9886794 4.712 0.194345 0589382 0.810147 0910425 0955728 0.977004

0.95 17 0.628 0.578741 0964906 0.995748 0999221 0999808 0.99994111 0.942 0.493486 0934262 0.989668 0997786 0999395 0.9998029 1.257 0.386566 0884078 0.977146 0994363 0998308 0.9994067 1.571 0.400057 0871118 0.971202 0992207 0997494 0.9990746 2.356 0.235906 0730075 0.914763 0970772 0988864 0.9953275 3.141 0.210884 066677 0.874695 0950566 0978959 0.9903535 3.942 0.114322 0510395 0.772728 0895273 0949855 0.9747934 4.712 0.194345 0589382 0.810147 0910425 0955728 0.977004

0.99 23 0.628 0.368485 0924445 0.989859 0998068 0999515 0.99985015 0.942 0.270600 0860178 0.974963 0994335 0998410 0.99947011 1.257 0.242353 0814696 0.95973 0989626 0996814 0.9988689 1.571 0.214900 0768031 0.941019 0983042 0994368 0.9978787 2.356 0.137884 0629726 0.871204 0953550 0981780 0.9922156 3.141 0.100809 0525516 0.798317 0914794 0962198 0.9821986 3.942 0.001457 0006660 0.014329 0023847 0034880 0.0472005 4.712 0.064841 0385076 0.666702 0826696 0908785 0.950547

risk, say , one may be interested in knowing the value of 0

that will ensure theproducers risk to be at most . We note that (2) can be written as

at =1

t0

0

0t0

(10)


which will denoted by at

0, for xed . Based on (10), the probability p =

F(at

), with F as in (5), may be obtained as function of

0, that is, p =

Fat

0. Then,

0is the smallest positive number for which p = Fat 0

satises the inequality

cx=0

(nx

)px1 pnx 1 (11)

For a given ASTLT plan n c t/0, at a specied condence level P, the

minimum values of 0

satisfying (8) were determined and are presented in Table 5,for xed = 1 and = 5.

3.4. Extensions and Approximations

Tables 35 can be generated for any other values of and , and also for anyother GBS distribution. A Mathematica program that carries out this computationis available from the authors upon request.

For T GBS t and xed and , if p = FTt is small and n islarge, the binomial probability can be approximated by the Poisson probability withparameter = np. So, (8) can be approximated as

cx=0

expxx! 1 P

(12)

Thus, a ASTLT plan n c t0 can be obtained based on this approximation.

Also, following a similar procedure as in Gupta (1962), it is possible to obtain anapproximate formula for n from (12).

Instead of the median, some other quantiles can also be used. The tablespresented in this article are directly usable in order to carry out ASTLT for meanor other quantiles of the GBS distribution or log-GBS distribution.

4. Description of Tables and Illustrative Examples

The numerical results for T GBS t, when = 1 and = 5, are presentedin Tables 35. Table 3 presents the minimum sample size necessary to assert thatthe median life exceeds a given value, 0, with probability P

and the correspondingacceptance number, c, using binomial probabilities. Table 4 presents the operatingcharacteristic values for the sampling plan n c t

0 for a given P when c = 2.

Finally, Table 5 presents the minimum ratios of true median life to specied medianlife for the acceptance of a lot with producers risk of 0.05. We shall now presenttwo illustrative examples.

4.1. Example 1

Suppose that the lifetime (T ) of a product follows a generalized BirnbaumSaundersdistribution. Specically, let T GBS t, with = 1 (which is close to IFRAmodel) and = 5. An experimenter wants to establish that the true unknownmedian life is at least 1,000 hours (0) with condence P

= 095, and that the life


Table 5Minimum ratio of true median life to specied median life for the acceptance of a

lot with producers risk of 0.05 (when = 1 and = 5)0

P c 0.628 0.942 1.257 1.571 2.356 3.141 3.972 4.712

0.75 0 7420 9699 10603 13252 13878 18502 23398 277561 3855 4353 5006 6257 7591 10121 8863 105152 2850 3457 3720 3989 4822 6429 8130 67103 2517 2828 3146 3485 4476 4813 6085 72194 2236 2472 2817 3186 3623 3903 4935 58545 2122 2376 2602 2700 3073 4096 4195 49766 1983 2193 2449 2607 3131 3581 3676 43617 1878 2057 2183 2325 2794 3199 3292 39068 1841 2035 2114 2293 2532 2903 3671 35539 177 1942 2056 2265 2618 2667 3372 327210 1714 1866 2008 2094 2424 2869 3128 3042

0.90 0 8967 11130 12942 16175 19874 26495 23397 277571 4600 5351 5809 7260 9384 10121 12798 151832 3444 4020 4613 5235 5982 6429 8130 96443 2928 3238 3774 3932 5226 5967 6085 72194 2634 2943 3299 3521 4231 4831 6108 58555 2381 2748 2991 3251 3585 4096 5180 49766 2255 2510 2775 3060 3536 4175 4529 53727 2159 2333 2614 2730 3153 3724 4045 47988 2046 2272 2489 2642 3157 3376 3671 43559 1990 2156 2388 2570 2893 3097 3916 400010 1912 2121 2306 2377 2677 3231 3628 3710

0.95 0 10229 12360 14851 16175 19874 26496 33506 397481 5027 5782 6511 7260 9384 12511 12798 151822 3704 4274 5002 5235 6972 7974 10084 96443 3113 3605 4056 4340 5226 5967 7546 72194 2846 3222 3518 3831 4778 5640 6109 72475 2558 2862 3171 3502 4050 4780 5180 61456 2403 2704 2927 3268 3910 4174 5279 53727 2287 2585 2745 3095 3486 4203 4710 47988 2197 2421 2604 2804 3439 3807 4268 43559 2125 2354 2492 2713 3152 3490 3916 464610 2036 2241 2400 2638 2915 3231 3628 4304

0.99 0 12275 15344 17949 20613 24257 32340 33505 397481 5945 6899 8247 8925 10888 12511 15821 187692 4383 5166 5704 6252 7850 9295 10084 119633 3610 4246 4810 5069 6507 6966 8810 89524 3229 3723 4117 4398 6508 6370 7133 84615 2931 3284 3667 3963 4876 5398 6044 71706 2721 3059 3349 3657 4259 5213 5961 62637 2564 2891 3229 3430 4093 4648 5316 55878 2442 2761 3033 3255 3707 4209 4814 50649 2343 2598 2877 3114 3630 4202 4414 523610 2263 2518 2749 2999 3359 3887 4512 4847


test will be terminated at t = 942 hours. Then, for an acceptance number c = 2, therequired n is found from Table 3 to be 11. If during 942 hours, no more than 2failures out of 11 are observed, then the experimenter can assert with condence 0.95that the median life is at least 1,000 hours. Instead of the binomial formula, if thePoisson approximation is used, the value of n would instead be 14. The values of npresented in Table 3 are indeed less than the corresponding values of n tabulated inof Gupta and Groll (1961) for gamma distribution with shape parameter 2 (whichis an IFR model), Kantam et al. (2001) for the log-logistic distribution, and Bakliziand El Masri (2004) for the BS distribution for = 1 (which is approximately anIFRA model).

For the sampling plan n = 11 c = 2 t/0 = 0942, the operating characteristicvalues from Table 4 are as follows:

02 4 6 8 10 12

t0

0.493486 0.934262 0.989668 0.997786 0.999395 0.999802

This simply means that if the true median life is twice the specied median life( 0

= 2), then the producers risk is about 0507, while it is about 0066 when thetrue median life is 4 times the specied median life. Table 4 can be used to get thevalue of

0for various choices of (c t

0) such that the producers risk may not exceed

005. For example, the value of 0

is 4274 for c = 2 t0

= 0942 and P = 095. Thismeans that the product should have a median life of 4274 times the specied medianlife in order for the lot to be accepted with probability 095.

4.2. Example 2

Consider a problem associated with software reliability provided by Wood (1996)and analyzed from the acceptance sampling viewpoint by Rosaiah and Kantam(2005). We consider the failure times in hours (T ) of the release of a software,which times correspond to the lifetimes from the starting of the execution of thesoftware until which the failure of the software is experienced. We assume that whilethe software is operating, the development of intangible cumulative degradationdeteriorates the performance of this software. Then, it is reasonable to suppose thatthe r.v. T follows a generalized BirnbaumSaunders distribution. Through T , it willbe possible to classify one unit of the software release like defective or non-defective.

Let the specied median life be 0 = 1,000 hours and the testing time be t =1,257 hours, which leads to the ratio t

0= 1257. Thus, for an acceptance number

c = 2 and condence level P = 095, the required n is found from Table 3 to be9; that is, in this case, the acceptance sampling plan from truncated life tests fromGBS(t distribution is n = 9 c = 2 t0 = 1257.

We consider the ordered sample of size n = 9 from T (in hours) (ti i = 1 9):519, 968, 1430, 1893, 2490, 3058, 3625, 4422, and 5218. Based on these data, wehave to decide whether to accept or reject the lot. We will accept the lot only if thenumber of failures before 1,257 hours is at most 2.

As the condence level is assured by this sampling plan only ifT GBS t, then we should study if it is reasonable to admit that the givensample comes from the generalized BirnbaumSaunders distribution. So, based on


Figure 1. PP plot and determination coefcient (R2) for the indicate distributions.

the proposal (1)(5) in Sec. 1, we t this model to the data. In order to estimatethe parameters of the assumed distribution, we will use the method of maximumlikelihood estimation for the parameters and based on the algorithm of EMtype. This t will then be compared with the classical BirnbaumSaunders (BS) andinverse Rayleigh (IR) models.

Figure 1 shows the probability versus probability (PP) plots of the data and thecorresponding PP determination coefcients for these models. Based on Fig. 1, wecan say the GBS and BS models t the data reasonably well, but the PP correlationsreveal that the GBS(t8) distribution ts the data better.

Since in the given sample of n = 9 observations, there are only c = 2 failures at519 and 968 hours before t = 1,250 hours, we shall accept the lot, assuring a medianlife 0 = 1,000 hours with a condence level of P = 095.

5. Concluding Remarks

In this article, we have developed acceptance sampling plans when the life test istruncated at a pre-xed time and the lifetimes of the test units follow a Student-tgeneralized BirnbaumSaunders distribution. We have shown in general that undersimilar conditions, in order to ensure a specied median life with a given condencelevel, the GBS(t) model results in smaller sample sizes than some other modelsused in acceptance sampling. We have also demonstrated for a real data set that theGBS(t) distribution ts the data better than the classical BirnbaumSaunders andinverse Rayleigh models.

Acknowledgments

This study was carried out with the support of research projects FONDECYT1050862, FANDES C-13955/10, and DIPUV 42-2004, Chile. This article waswritten partially during the time that Dr. Vctor Leiva was doing a PostdoctoralFellow in the Department of Mathematics and Statistics, McMaster University,Canada. He is especially grateful with the McMaster University, Canada, for itssupport.

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