[[abstract]]Considering the producer and consumer risks, the paper develops acceptance sampling procedures under the progressively interval-censored test with intermittent inspections for the exponential lifetime model. The proposed approach allows removing surviving items during the life test such that some extreme lifetimes can be sought, or the test facilities can be freed up for other tests. A reduction in testing effort and administrative convenience can be achieved by employing the proposed approach. One example is introduced for illustration.[[conferencetype]]國際[[conferencedate]]20080618~20080620[[conferencelocation]]Dalian, Chin

Tsai, Tzong-ru

Considering the producer and consumer risks, the paper develops acceptance sampling procedures under the progressively interval-censored test with intermittent inspections for the exponential lifetime model. The proposed approach allows removing surviving items during the life test such that some extreme lifetimes can be sought, or the test facilities can be freed up for other tests. A reduction in testing effort and administrative convenience can be achieved by employing the proposed approach. One example is introduced for illustration.國際20080618~20080620Dalian, Chin

Tamkang University Institutional Repository

Progressively Interval-censored Life Test with Acceptance SamplingTzong-Ru Tsai1∗, Kuang-Nan Lin2, Jyun-You Chiang3, Hsaio-Ching Fan4124Department of Statistics, Tamkang University, Tamsui, Taipei 251, Taiwan R.O.C.3Graduate Institute of Management Sciences, Tamkang University, Tamsui, Taipei 251, Taiwan R.O.C.Corresponding author: T.-R. Tsai, trtsai@stat.tku.edu.twAbstractConsidering the producer and consumer risks, the paperdevelops acceptance sampling procedures under the pro-gressively interval-censored test with intermittent inspec-tions for the exponential lifetime model. The proposed ap-proach allows removing surviving items during the life testsuch that some extreme lifetimes can be sought, or the testfacilities can be freed up for other tests. A reduction in test-ing effort and administrative convenience can be achievedby employing the proposed approach. One example is intro-duced for illustration.1 IntroductionAcceptance sampling is a set of rules concerning withinspection and decision making regarding items. In life-testing applications, the quality characteristic of an item of-ten is measured by its lifetime. Acceptance sampling planused to determine the acceptability of items with respect totheir lifetimes are also called the life test plan (LTP). Sinceengineering technologies have highly developed nowadays,most products have been designed with high reliabilities.Thus, how to develop acceptance sampling procedures forhigh reliability items becomes an important problem tomanufacturers.In life-testing applications, it may take a long time tocollect complete lifetime data. For saving test time, LTPsoften are developed under the type I censoring or the type IIcensoring schemes. In a type I censoring scheme, life testsare run over a fixed time period in such a way that an item’slifetime will be known exactly only if it is less than or equalto some predetermined values. Alternatively, in a type IIcensoring scheme, the experimenter terminates the life testwhen the r smallest observations in a random sample of nitems are observed.Traditionally, a LTP consists of the sample size requiredin the life test and the corresponding critical value such thatthe quality personnel can make a decision to the testing lotof items based on the critical value. For example, the LTPcan be designed to protect both the produce and consumersuch that the produce risk and consumer risk are lower thanor equal to the required levels. The produce risk denotesthe probability that a good lot of items will be rejected.The consumer risk denotes the probability of accepting alot of items with poor quality. Based on the producer andconsumer risks, some LTPs have been developed. Epsteinand Sobel [8] constructed LTPs for exponential distributionwith type II censoring. Epstein [7] proposed a hybrid LTPwhich combines the type I censoring and the type II cen-soring. Jeong and Yum [10] developed LTPs for exponen-tial distribution with type I censoring. Kim and Yum [11]did comparisons of different designs of the exponential LTPwith intermittent inspections. Wu and Tsai [18] developedthe design of truncated LTP for Birnbaum-Saunders distri-bution. Tsai and Wu [14] proposed the design of truncatedLTP for generalized Rayleigh distribution.The type I and type II censoring schemes, however, donot allow removing surviving items at the times other thanthe termination time of the life test. This allowance, how-ever, may be desirable when a compromise between reduc-ing test time and an expectation of some extreme lifetimesin life test can be sought. This reason motivates the qual-ity personnel to adopt a progressively censoring scheme, forexample, the progressively type I censoring scheme and theprogressively type II censoring scheme. The progressivelytype I censoring scheme allows removing surviving itemsat some fixed times before the termination time of life test,and the progressively type II censoring scheme allows re-moving surviving items at the time of each failure of ther − 1 smallest observations are observed if r failures arerequired among n test items. A comprehensive referenceregarding the progressive censoring tests, its applicationsand mathematical results can be found in Balakrishnan andAggaarwala [3]. The parameter estimations of the expo-nential, lognormal and Weibull lifetime models with pro-gressive type I censoring have been discussed by Cohen [6],and Wong [17], respectively. The problems of simulation,estimation, calculation of moments and the construction ofThe 3rd Intetnational Conference on Innovative Computing Information and Control (ICICIC'08) 978-0-7695-3161-8/08 $25.00 © 2008 IEEEa LTP with progressive type II censoring have been dis-cussed by Aggarwala and Balakrishnan [2], Balakrishnanand Sandhu [4], Balakrishnan and Saw [5], Tse and Yuen[13], Viverps and Balakrishnan [16] and Yuen and Tse [19],respectively.2 MotivationIt is easier to develop the statistical inferences with typeII censoring data than using the type I censoring data. Butthe termination time of life test under a type II censoringscheme is randomly and cannot be predetermined. More-over, the quality personnel may inspect the test items onlyat some specific times during the life test so that the exactfailure times cannot be observed. These reasons encour-age quality personnel to conduct a life test with an interval-censored scheme which only records the failure numbers insome predetermined time intervals and not record the exactlifetimes of items continuously. Based on the administra-tive convenience and a compromise between reducing testtime and the allowance to remove surviving items during thelife test, the study is thus motivated to develop LTPs undera progressively interval-censored scheme with intermittentinspections. Basically, the construction of the proposed LTPis easy for practitioners.3 Progressively Interval-censored TestAssume that the lifetimes of items Y1, Y2, · · · are in-dependent and identically distributed as an exponentiallydistributed with the following cumulative density function(c.d.f.)F (y|η) = 1− exp(−yη), y > 0, η > 0, (1)where η is the scale parameter and the mean lifetime ofitems. Assume that a lot of items taken from (1) are sub-mitted to an inspection of acceptance sampling. The lot isaccepted if the mean lifetime η is longer than or equal toη0, and rejected if the mean lifetime reduces to ηa(< η0)or less, where η0 and ηa are two predetermined level. Inthe development of the life test, we can transform the life-time data by letting Ti = Yi/η0, i = 1, 2, · · · so that thetransformed data do not depend on the measuring scale. Letθ = η/η0, it can be shown that T1, T2, · · · are independentand identically exponentially distributed with parameter θ.The progressively interval-censored test can be con-ducted as follows: Assume that m(≥ 2) inspection times0 < τ1 < τ2 < · · · < τm < ∞ are predetermined and nitems are drawn randomly from the lot and put on a life testat the initial time 0. At the time τ1, X1 failure items in theinterval (0, τ1] are recorded, and R1 of the n −X1 surviv-ing items are selected randomly and removed. Continuingon the test, at the time τ2, X2 failure items in the interval(τ1, τ2] are recorded, and R2 of the n − X1 − X2 − R1surviving items are selected randomly and removed. Fi-nally, at the termination time τm, Xm failure items in theinterval (τm−1, τm] are recorded, and all surviving num-bers Rm = n−∑mj=1 Xj −∑m−1j=1 Rj are removed. Thenthe test is stopped. In the life test, either the proposed val-ues of R1, R2, · · · , Rm−1, or probabilities of units removedp1, p2, · · · , pm−1 are predetermined such that pm = 1 andRi = [(n −∑ij=1 Xj −∑i−1j=1 Rj)pi], i = 1, 2, · · · ,m,where [s] is the largest integer smaller than or equal to s.4 Ordinary and Approximate Life Test PlansAssume that a random sample of size n are drawn froma lot and put on a progressively interval-censored test withgiven numbers of removals R = (R1, , R2, · · · , Rm). Thefailure numbers in the given time intervals are collectedand denoted by X = (X1, X2, · · · , Xm), respectively.Let B(a, b) denote the binomial distribution with the num-ber of individuals a and the probability of success b. Letν1 = n, δ1(θ) = F (τ1|θ), νi = n −∑i−1j=1(Xj + Rj)and δi(θ) =F (τi|θ)−F (τi−1|θ)1−F (τi−1|θ)= 1 − exp(− τi−τi−1θ),i = 2, 3, · · · ,m. It can be shown that X1 ∼ B(ν1, δ1(θ))and for i = 2, 3, · · · ,m, Xi|Xi−1, ..., X1, Ri−1, ..., R1 ∼B (νi, δi(θ)).The statistical hypotheses regarding the acceptance sam-pling of the life test can be described as H0 : θ = 1 vs.H1 : θ = θa(< 1). The θa is so-called the discriminationratio. In particular, the life test is conducted with intermit-tent inspections, that is, m inspections are equally-spacedsuch that the length of the i-th time interval τi − τi−1 = τand δi(θ) = δ(θ) = 1 − exp(−τ/θ), i = 1, 2, · · · ,m. IfX1 = n then the log-likelihood is maximized when θ ap-proaches to zero, otherwise, if Rm = n, then log-likelihoodis maximized when θ approaches to infinity, otherwise themaximum likelihood (ML) estimator of θ can be obtained.According to the computation procedure of Aggarwala [1],the likelihood estimator of θ can be found and denoted byθˆ (see Tsai et al.[15]). The producer risk and the consumerrisk are defined respectively as PR = P (θˆ < c|θ = 1) =∑ReP (x|R, θ = 1) and PC = P (θˆ ≥ c|θ = θa) =1 −∑Re P (x|R, θ = θa), where c is a constant called thecritical value, Re = {x|∑mi=1(xi + Ri) = n, θˆ < c}, andP (x|R, θ) is the joint probability of X = x. Based on theML estimate of θˆ, the ordinary LTP (n, c) can be developedsuch that the producer and consumer risks as follows aresatisfied:PR ≤ α and 1− PC ≥ 1− β. (2)Assume that the number of removals are given by Ri =[(n−∑ij=1 Xj−∑i−1j=1 Rj)pi], i = 1, 2, · · · ,m−1, whereThe 3rd Intetnational Conference on Innovative Computing Information and Control (ICICIC'08) 978-0-7695-3161-8/08 $25.00 © 2008 IEEER0 = 0. A searching procedure with the following steps issuggested to establish the ordinary LTP:Step 1: Specify the values of m, τ , α, β, θa andp1, p2, · · · , pm.Step 2: Let n = 3.Step 3: Generate all possible combinations of x and R basedon the process of X1 ∼ B(ν1, δ1(θ)) and fori = 2, 3, · · · ,m, Xi|Xi−1, ..., X1, Ri−1, ..., R1 ∼B (νi, δi(θ)), and compute the ML estimate of θ foreach combination of x and R, and denoted by θˆ.Step 4: The sets of {θˆ, P (x|R, θ = 1)} and the{θˆ, P (x|R, θ = θa)} for all possible combina-tions of x constitute the p.d.f.’s of θˆ under θ = 1and θ = θa, respectively. Hence, the c.d.f.’s of θˆunder θ = 1 and θ = θa can be determined byaccumulating the joint probabilities of P (x|R, θ = 1)and P (x|R, θ = θa), respectively.Step 5: If there exist c’s for which equations in (2) are satisfied,then select the largest one among those c’s and stop.The desired ordinary LTP is determined. Otherwise,go to Step 6.Step 6: Let n = n + 1, go to Step 3.The ordinary LTPs are exact, however, the method mayfail due to a memory overflow error especially for largem and n. The computation is complicated and timeconsuming. For get over this difficulty, an approxi-mate method is proposed to find the approximate LTPs.Based on asymptotic theory of ML estimator, we canapproximate the sampling distribution of θˆ by a nor-mal distribution with mean θ and variance I−1(θ), whereI(θ) is the Fisher information (see Hogg and Craig[9]). Tsai et al. [15] prove the following results: Letφ1(θ) = δ(θ), and for i = 2, 3, · · · ,m, let φi(θ) =δ(θ)[1−∑i−1j=1 pj(1− δ(θ))2i−j−1 ∏i−1h=1(1− ph)],G1(θ) =[τ(1− δ(θ))θ3δ(θ)] [τθδ(θ)− 2] m∑i=1φi(θ),G2(θ) =2τθ3m∑i=1(m− i + 1)φi(θ);if m = 2, let G3(θ) = 2τθ3 [2− (1− δ(θ))p1]; if m = 3, letG3(θ) =2τθ3[m− (m− 1)(1− δ(θ))p1−m−1∑i=2(m− i)pi(1− δ(θ))ii−1∏j=1(1− pj) .Frequency0 2000 4000 6000 8000 10000051015Figure 1. Histogram of the number of cycles to failure data.Let G(θ) = G1(θ) + G2(θ) + G3(θ). It can be shownthat I(θ) ' nG(θ), and the required sample size can bedetermined byn '[z1−β/√G(θa) + z1−α/√G(1)1− θa]2,c = 1− z1−α√n√G(1)or c = θa +z1−β√n√G(θa).5 Illustrative ExampleA sample of the number of cycles to failure for a group of60 electrical appliances in a life test are reported on P.112of Lawless [12]. Figure 1 indicates that the histogram ofthe data is exponentially decayed so that it is reasonable toroughly characterize the data by an exponential distribution.Assume that we treat the data as a random sample drawnfrom a lot of such items and put on a progressively interval-censored test with two equally-spaced inspection times andthe termination time 450. Let the produce risk α = 0.05, theconsumer risk β = 0.1, η0 = 4500, ηa = 1500, p1 = 0.5and p2 = 1. It can be shown that θ0 = 1, θa = 1/3,τ = 1/20. At the time 225, there are 10 failure items arefound in the interval (0, 225] or X1 = 10, and R1 = (60−10) × 0.5 = 25 surviving items are selected randomly andremoved. Continuing on the test, at the time 450, we finda item fails in the interval (225, 450] or X2 = 1, and allsurviving items R2 = 60 − 10 − 25 = 25 are removed.Then the test is stopped. The ordinary LTP can be foundas (n, c) = (60, 0.5183). The ML estimate of θ can bedetermined as θˆ = 0.3608. Since θˆ < 0.5183, we reject thelot.6 ConclusionsIn this paper, the design of progressively interval-censored sampling plans with equally-spaced inspectiontimes is developed based on the ordinary and approximateThe 3rd Intetnational Conference on Innovative Computing Information and Control (ICICIC'08) 978-0-7695-3161-8/08 $25.00 © 2008 IEEEmethods. the use of the proposed method is illustrated by anexample. The ordinary life test plans are exact, however, thecomputation is complicated and time consuming if the num-ber of inspection and the sample size are large. Moreover,the search procedure may fail due to a memory overflowerror on computer. If the computer facilities are excellentfor practitioners, the ordinary life test plans are suggested.Otherwise, the approximate life test plans are suggested toreplace the ordinary life test plans.The proposed sampling plans can be applied to theWeibull lifetime model with known shape parameter. Thepractitioners can transform the Weibull lifetime data into theexponential data, then use the proposed approach. Extend-ing the proposed study to the case of two-parameter lifetimemodels may be a fruitful area of future research.References[1] Aggarwala, R. Progressive interval censoring: somemathematical results with applications to inference.Commun. Statist.-Theory Meth., 30:1921-1935, 2001.[2] Aggarwala, R. and Balakrishnan, N. Some propertiesof progressive censored order statistics from arbitraryand uniform distributions with applications to infer-ence and simulation. J. Statist. Plan. Infer., 70:35-49,1998.[3] Balakrishnan, N. and Aggarwala, R. Progres-sive censoring-theroy, methods, and applications,Birkha¨user, Boston, 2000.[4] Balakrishnan, N. and Sandhu, R.A. A simple simu-lational algorithm for generating progressive Type-IIcensored samples. Amer. Statist., 49:229-230, 1995.[5] Balakrishnan, U. and Saw, S.L.C. Reliability samplingplans for the two-parameter exponential distributionunder progressive censoring. J. Appli. Statist., 25:707-714, 1998.[6] Cohen, A.C. Progressively censored samples in lifetesting. Technometrics, 5:327-329, 1998.[7] Epstein, B. Truncated life tests in the exponential case.Ann. Math. Statist., 25:555-564, 1954.[8] Epstein, B. and Sobel, M. Life testing. J. Amer. Statist.Assoc., 48:486-502, 1953.[9] Hogg, R.V. and Craig, A.T. Intro. Math. Statist., 5ed,Prentice Hall, 1995.[10] Jeong, H. S. and Yum, B. J. Type-I censored life testplans in the exponential case. Commun. Statist.-Simul.Comput., 24:187-205, 1995.[11] Kim, S.-H. and Yum, B.-J. Comparisons of expo-nential life test plans with intermittent inspections. J.Qual. Technol., 32:217-230, 2000.[12] Lawless, J. F. Statistical Models and Methods for Life-time Data, 2nd, John Wiley, 2003.[13] Tse, S.-K. and Yuen, H.-K. Expected experimenttimes for the Weibull distribution under progressivecensoring with random removals. J. Appli. Statist.,25:75-83, 1998.[14] Tsai, T.-R. and Wu, S.-J. Acceptance sampling basedon truncated life tests for generalized Rayleigh distri-bution. J. Appli. Statist., 33:595-600, 2006.[15] Tsai, T.-R., Chiang, J.-Y. and Fan H.-C. Progressivelyinterval-censored sampling plans with intermittent in-spections. Technical Report-01, Department of Statis-tics, Tamkang University, 2008.[16] Viveros, R. and Balakrishnan, N. Interval estima-tion of life characterisitcs from progressively censoreddata. Technometrics, 36:84-91, 1994.[17] Wong, J. Y. Simultaneously estimating the threeWeibull parameters from progressively censored sam-ples. Microelectronics Reliab., 33:2217-2224, 1993.[18] Wu, C.-J. and Tsai, T.-R. Acceptance sampling plansfor Birnbaum-Saunders distribution under truncatedlife tests. Int. J. Reliab. Qual. Saf. Eng., 12:507-519,2005.[19] Yuen, H.-K. and Tse, S.-K. Parameters estimationfor Weibull distributed lifetimes under progressivecensoring with random removals. J. Statist. Comput.Simul., 55:57-71, 1996.The 3rd Intetnational Conference on Innovative Computing Information and Control (ICICIC'08) 978-0-7695-3161-8/08 $25.00 © 2008 IEEE

Progressively interval-censored life test with acceptance sampling

Tzong-Ru Tsai

Kuang-Nan Lin

Jyun-You Chiang

Hsaio-Ching Fan

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Progressively Interval-Censored Life Test with Acceptance Sampling

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Progressively interval-censored life test with acceptance sampling

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