加速寿命试验

Assessing the Lifetime Performance Index of Exponential Products With Step-Stress AcceleratedLife-Testing DataHsiu-Mei Lee,Jong-Wuu Wu,and Chia-Ling LeiAbstract—Lifetime performance assessment has been crucial to the manufacturing industry.In practice,a lifetime performance index is used to measure the larger-the-better type quality characteristics.Accelerated life test(ALT)has often been used to yield information quickly so that the life distribution of products can be estimated.This study constructs a maximum likelihood estimator(MLE)of for exponential products based on type II right censored data from the step-stress accelerated life test (SSALT).The MLE of is then utilized to develop the hypothesis testing procedure with the given lower specification limit.This new testing procedure can be easily applied to assess whether the lifetime of products meets the requirements.Finally,we give two examples to explicate the proposed testing procedures.Index Terms—Hypothesis testing procedure,lifetime per-formance index,maximum likelihood estimator,step-stress accelerated life test,type II right censored data.A CRONYMALT accelerated life testSSALT step-stress accelerated life testPDF probability density functionCDF cumulative distribution functionMLE maximum likelihood estimatorN OTATIONmean of exponential distributionmean lifetimelifetime standard deviationlower specification limitlifetime performance indexML E ofManuscript received September24,2011;revised February20,2012and June 28,2012;accepted October10,2012.Date of publication February05,2013; date of current version February27,2013.Associate Editor:R.H.Yeh.H.-M.Lee is with the Department of Statistics,Tamkang University,New Taipei City,Taiwan(e-mail:079045@.tw).J.-W.Wu is with the Department of Applied Mathematics,National Chiayi University,Chiayi City,Taiwan(e-mail:jwwu@.tw).C.-L.Lei is with the Department of Surgery,Ditmanson Medical Foundation Chia-Yi Christian Hospital,Taiwan(e-mail:11371@.tw).Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/TR.2013.2241197design stress levelstress levels,,2total number of test unitsnumber of failures atnumber of failure-censored observations:th observed failure time,mean life of a test unit atMLE ofthe mean life of a test unit at,parameters for log-linear stress-level model,MLE of,CDF of WPDF of Chi-square distributioncritical valuec target valueN(0,1)standard normal distributionapproximately normal distributionapproximately bivariate normal distributionth percentile of W distributionth percentile of standard normaldistributionpower functionapproximate power functionI.I NTRODUCTIONP ROCESS capability analysis is an effective means of measuring process performance and potential capability. Various methods have been developed for assessing the quality of a product.Process capability analysis has several benefits: continuously monitoring the process quality through process capability indices(PCIs)to ensure the products manufactured are meeting to the required level;supplying information on product design and process quality improvement for engineers and designer;and providing the basis for reducing the cost0018-9529/$31.00©2013IEEEof product failures.Process capability indices,including, ,,and,are widely used to measure process potential and performance.These four indices measure the target-the-better type quality characteristics with bilateral tol-erances.Montgomery[13],and Kane[7]provided the PCIs, such as,,and,for unilateral tolerances,where denotes the index measuring the smaller-the-better type quality characteristics,and and represent the indices that measure the larger-the-better type quality characteristics. Because the lifetime of electronic components exhibits the larger-the-better quality characteristic of time orientation, Montgomery[13]recommends the lifetime performance index for evaluating the lifetime performance of electronic components.Tong et al.[16]construct a uniformly minimum variance unbiased estimator(UMVUE)of based on an exponential distribution with the complete sample,and then the UMVUE of can be used to develop the hypothesis testing procedure.Purchasers can then employ the testing procedure to determine whether the lifetime of electronic components adheres to the required level.Through the same procedure, manufacturers can also enhance the process capability.They also noted that if the quality characteristic possesses a specific non-normal distribution(such as exponential),then the PCI es-timators obtained by percentile approaches are biased.Several authors,including Hong et al.[5],[6],Lee et al.[8]–[10],and Wu et al.[22],[23],have discussed the lifetime performance index for assessing the lifetime performance of products based on censored data.All of the papers noted above have made inferences con-cerning the lifetime performance index of products based on life-testing experiments under the normal operating condition. Needless to say,life-testing under the normal operating condi-tion is more reliable and accurate.However,in many situations, step-stress accelerated life test(SSALT)can also obtain infor-mation about the reliability of a product in a short time,and re-duce the cost.The problem of modeling the data of SSALT and making inferences from such data have been studied by many authors such as Bai et al.[1],Balakrishnan et al.[2],Ma and Meeker[11],Miller and Nelson[12],Nelson[14],Wang[17], Watkins[21],Wang and Fei[20],Xiong[24],and Xiong et al.[25].Therefore,our study will make an inference for the life-time performance index of products from step-stress accel-erated life-testing data.There are many ways of applying stress to the test units.One employs a step-stress scheme which allows the stress setting of a unit to be changed at pre-specified times; another changes the stress when afixed number of failed units occur.The former is called“time-step-stress test,”and the latter is called“failure-step-stress test.”Compared with the failure-step-stress ALT model,the time-step-stress ALT model has the following problems.1)When there is no observation under some particular stress-levels,the MLEs for the parameters corresponding to these stress levels do not exist.2)In addition,it is also quite complicated to develop the statistical inference for the parameters.Many researchers have been aware of this issue.Therefore,a study of the failure-step-stress ALT model has also been considered,such as Chung et al.[3],Teng and Yeo[15],and Wang[17],[18].In this paper, we present a MLE of based on an exponential distribution with a simple failure-step-stress accelerated life-testing sample, and the MLE of is then used to develop the hypothesis testing procedure.The hypothesis testing procedure can be em-ployed to determine whether the lifetime performance meets the required level.The rest of this paper is organized as follows.In Section II, we discuss the relationship between the lifetime performance index and the conforming rate of products.Section III provides some basic assumptions.In Section IV,we present the MLE of ,and its main properties.In Section V,we establish the hy-pothesis testing procedure for the lifetime performance index, and we compare the exact and approximate power of this hy-pothesis test under a given condition.Two examples are applied to explicate the hypothesis testing procedure in Section VI.Fi-nally,some concluding comments are given in Section VII.II.T HE L IFETIME P ERFORMANCE I NDEX Suppose that the lifetime of products can be modeled by an exponential distribution.Let Y denote the lifetime of products, and Y has an exponential distribution with the probability den-sity function(PDF),and cumulative distribution function(CDF) respectively as(1)and(2)The lifetime of products is a larger-the-better type quality char-acteristic.Hence,the lifetime is generally required to exceed the unit time to be bothfinancially profitable and customer-sat-isfactory.Montgomery[13]suggests a capability index to measure the larger-the-better quality characteristics.is de-fined as(3)In(3),denotes the mean lifetime,represents the lifetime standard deviation,and L indicates the lower specification limit. To assess the lifetime performance of products,can be defined as the lifetime performance index.If has an exponen-tial distribution with the probability density function as given in (1),then the lifetime performance index can be reduced to(4)The larger the mean,the larger the lifetime performance index .Therefore,the lifetime performance index reasonably and accurately represents the lifetime performance of products. Moreover,if the lifetime of a product exceeds the lower speci-fication limit(i.e.),then the product will be labeled as conforming.The ratio of conforming products is known as the conforming rate,and can be defined as(5)TABLE IT HE L IFETIME P ERFORMANCE INDEXVS .THEC ONFORMING R ATEObviously,a strictly increasing relationship exists between the conforming rate and the lifetime performance index.Hence,Table I lists variousvalues,and the corresponding conforming rates .Forvalues not listed in Table I,the conforming rate can be obtained through interpolation.The con-forming rate can also be calculated by dividing the number of conforming products by the total number of products sampled.To accurately estimate the conforming rate,the sample size must be large.Therefore,utilizing the one-to-one relationship be-tween and ,the lifetime performance index can be a flex-ible,effective tool,not only for evaluating product quality but also for estimating the conforming rate .III.B ASIC A SSUMPTIONStatistical inference for the exponential step-stress ALT usu-ally depends on the following assumptions.1)For any stress level ,the lifetime distribution of a test unit is exponential with CDF:where is the mean life of a test unit at stress level .2)At a stress level ,the mean life of a test unit is a log-linear function of stress that can expressed as(6)where and are unknown parameters.3)A cumulative exposure model holds that the remaining life of a test product depends only on the current cumulative exposure and stress regardless of how the probability is accumulated.IV .MLE OF L IFETIME P ERFORMANCE I NDEXConsider the following simple step-stress accelerated life test with type II censoring.Suppose that all of the n test units are initially placed under low stress ;and when units have failed,the stress level is changed to .The test continues until failures occur at stress level ,and the remaining unitsare censored.Lettype II right a ALT.Based on assumptions 1and 3given above,the likelihood function of is given as(7)whereNow,through (7),and Assumption 2,the log-likelihood func-tion of can be rewritten as(8)We can obtain the MLEs for and,which can be expressedasandrespectively.The Fisher information matrix can be obtained by taking the expectation of the negative of the second partial derivatives of with respect to and .See the Appendix for details.Then the Fisher information matrix isBy the general asymptotic theory of MLEs,we obtain that(9)Based on Assumption2,and the invariance of the MLE(see Zehna[26]),the MLE of the mean life at a designed stress level is given as(10) And the MLE of is given as.By using(9)and(10),we can obtain the result as where.See the Appendix for more information.Moreover,we can express the mean life,and MLE re-spectively as(11) and(12) Let.By using(11)and(12),we obtainBy using Theorem1of Wang[19],the CDF of is given as(13) where,is the PDF of.V.T ESTING P ROCEDURE FOR THE L IFETIME P ERFORMANCEI NDEXIn this section,we construct a statistical testing procedure to assess whether the lifetime performance index adheres to the re-quired level.From(5),the performance index is available to determine the conforming rate of products.To assess the life-time performance of products,it is feasible to construct a testing procedure for the index.If the required index value of per-formance is larger than c,where c denotes the target value,then the null hypothesis and the alternative hypothesis can be ing as the test statistic, the rejected region can be express as.Given the specified significance level,the critical value can be cal-culated as(14)Because the CDF of is given as(13),and let be the th percentile of,then from(14)we obtain Thus,the following critical value can be derived:(15) Table II lists the95th percentiles of the distribution for the different and under and1.5by using the software Compaq Visual Fortran[4].Moreover,we use the asymptotic normality of the MLE,the critical value can be calculated asThus,the approximate critical value is given as(16) where is the th percentile of.The proposed testing procedure about can be structured as follows.Step1)Determine the lower lifetime limit,and the perfor-mance index value.Then the testing null hypoth-esis,and the alternative hypothesiscan be established.Step2)Specifying a significance level,and then obtaining the critical value.Step3)Through(12),calculate the value of.Step4)Calculate the value of the test statisticStep5)The decision rule of the statistical test is as follows.If,we may conclude that the lifetimeperformance index of products meets the requiredlevel.Based on the proposed testing procedure,the lifetime perfor-mance of products can be easily assessed.Moreover,the power of this statistic test is the probability of correctly rejecting a false null hypothesis.The power of this statistical test can be calcu-lated asTABLE II95TH P ERCENTILES OFD ISTRIBUTIONFOR,1.5whereis the CDF of.Moreover,if we construct the testing procedure based on the asymptotic normality of the MLE,the approximate power of the test can be calculated as (see the equation at the bottom of the page).The results of and are shown in Tables III and IV for given ,,,0.7,0.8,0.9,and,(10,20),(10,30),(20,10),(20,20),(20,30),(30,10),(30,20),(30,30).From Table III,Table IV,and based on ,and ,the following points can be drawn.(a)With fixed and ,as increases,the exact powerand approximate power also increase.(b)With fixed and ,as increases,the exact powerand approximate power also increase.TABLE IIIT HE V ALUES OF ,ANDFOR THE D IFFERENT,AND U NDER ,ANDTABLE IV T HE V ALUES OF ,ANDFOR THED IFFERENT,AND U NDER ,ANDTABLE VT HE D ATA OF F AILURE T IMES U NDER THE D IFFERENT A CCELERATED T EMPERATURE LEVELS(c)With fixed and ,as increases,the exact powerand approximate power also increase.(d)With ,0.8,and 0.9,the magnitudes of the dif-ference between the power and the approximatepower for small and are larger than for large and .(e)When ,,and are larger,the approximate poweris closer to the exact power .From (13)and (15),it is clear that the computation of the critical value based on the distribution is very dif ficult.When,and ,the approximate power is closerto the exact power .Therefore,we can use the testing procedure about based on the standard normal distribution with ,and .VI.N UMERICAL E XAMPLESIn this section,we apply the hypothesis testing procedure toboth a practical data set,and a simulated data set.Example 1:The first example is drawn from Section IV of Wang and Fei [20],and its purpose is to obtain the reliability indices of a kind of electronic component at the normal temper-ature level of .Now,items from a batch of products are randomly selected for the simple ALT model .The accelerated tem-perature levels are ,and respectively.Their failure times are presented in Table V.The data has been checked by Wang [18],and the lifetime distribution of test units at level is exponential.TABLE VIS IMULATED D ATA U NDER THE D IFFERENT S TRESS LEVELSThen,we restate the proposed testing procedure about as follows.Step 1)The lower lifetime limit is assumed to be 3000,and the conforming rate of products is required to exceed 80%.Referring to Table I,the value is required to exceed 0.80.Thus,the performance index value is set to .The testing hypothesisv.s.is constructed.Step 2)Specifying a signi ficance level ,and ob-taining a 95th percentile from(13)with ,,and ,then we obtain the critical valueby (15).Step 3)Using (12),we obtain thatStep 4)Calculate the value of the test statisticStep 5)Because of ,wereject the null hypothesis .Thus,we conclude that the lifetime performance index of the electronic component meets the required level.Alternatively,if we apply the normal distribution to this case,the approximate critical value is obtained using (16).Because of ,the null hy-pothesis is also rejected.Example 2:A sample of 50failure times are gener-ated from the simple SSALT model that uses two stress levels:,and .At the stress level ,the first 10failure times are observed ;at the stress level ,20failure times are observed;the remaining 20units are censored.The designed stress level is assumed to be .The model parameters are chosen to be ,and .The simulated data are presented in Table VI.Then,we restate the proposed testing procedure about as follows.Step 1)The lower lifetime limit is assumed to be 20.17,and the conforming rate of products is required to exceed 86%.Referring to Table I,the value is required to exceed 0.85.Thus,the performanceindex value is set to .The testing hypothesisvs.is constructed.Step 2)Specifying a signi ficance level ,and ob-taining a 95th percentile fromTable II with ,,and ,then we obtain the critical value from (15).Step 3)Using (12),we obtainStep 4)Calculating the value of the test statistic,Step 5)Because ,we reject tothe null hypothesis .Thus,we con-clude that the lifetime performance index of prod-ucts meets the required level.Alternatively,if we apply the normal distribution assumption to this case,the approximate critical value is ob-tained by (16).Because ,the null hypothesis is also rejected.In these two examples,the two methods applied to the data have consistent results.VII.C ONCLUSIONSeveral authors have employed the lifetime performance index to assess the lifetime performance of products by non-accelerated life-testing data.In many situations,accel-erated life tests may be inevitable due to time limitation or cost restriction.In this paper,we have considered the case of the simple SSALT with type II censoring,and constructed the MLE of .The MLE of is then utilized to develop the hypothesis testing procedure with the given lower speci fica-tion limit .The proposed testing procedure about can be easily applied,and can effectively evaluate whether the lifetime of products meets the requirements.We have observed that the computation of the approximate critical value is less complicated than the exact one from the W distribution.Hence,our recommendation is to use the testing procedure about based on a distribution whenever possible,and the testing procedure about based on a normal distribution in the case of a larger number of observed failures when the computation of exact critical values becomes dif ficult.The hypotheses testing procedure about is similar,when we use the proposed method on multiple step-stress ALT data.However,the MLEs need to be extracted numerically.Applyingthe least squares approach to estimate the parameters in a mul-tiple SSALT model,exploring other SSALT models,or com-paring these models can be an interesting area for future re-search.A PPENDIXBy the log-likelihood function of,Let the Fisher information matrix bewhereBy Assumption2,and Theorem3of Wang[18],we obtain that is-independent of,and.Hence,.following equations can be derived.The Fisher information matrix is rewritten asThe Fisher information can be inverted to get the asymptotic variance-covariance matrix of the MLEs asThus we can 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one-parameterexponential lifetime model with the upper record values,”Proc.Inst.Mechan.Eng.,Part B:J.Eng.Manufacture,vol.222,pp.1739–1759,2008.[24]C.Xiong,“Inference on a simple step-stress model with Type-II cen-sored exponential data,”IEEE Trans.Rel.,vol.47,pp.142–146,1998.[25]C.Xiong,K.Zhu,and M.Ji,“Analysis of a simple step-stress life testwith a random stress-change time,”IEEE Trans.Rel.,vol.55,no.1,pp.67–74,2006.[26]P.W.Zehna,“Invariance of maximum likelihood estimation,”Annal.Math.Statis.,vol.37,p.744,1966.Hsiu-Mei Lee received the Ph.D.degree in management science from Tamkang University,New Taipei City,Taiwan.She is currently an Associate Professor of the Department of Statistics at Tamkang University.Her articles have appeared in Applied Mathematics and Computation,International Journal of Systems Science,International Journal of Reliability,Quality and Safety Engineering,Mathematics and Computers in Simulation,and ICIC Express Letters,Part B:Applications.Her current re-search interests include quality control,and statistical applications in reliability. Jong-Wuu Wu received the Ph.D.degree in statistics from the National Central University,Zhongli City,Taiwan,in1992.He is a Professor of the Department of Applied Mathematics at National Chiayi University.His research interests include generalized linear models, statistical inference,reliability analysis,inventory theory,quality control,and fuzzy theory.He has published over100journal articles.His articles have appeared in a wide variety of journals,including Annals of the Institute of Statistical Mathematics,Applied Mathematics and Computation,Applied Mathematical Modeling,Applied Soft Computing,Biometrical Journal, C ommunications in Statistics–Theory and Methods,Fuzzy Sets and Systems, the IEEE T RANSACTIONS ON R ELIABILITY,Information Sciences,International Journal of Production Economics,International Journal of Systems Science, Statistical Papers,Statistics,The Australian and New Zealand Journal of Statistics,etc.Chia-Ling Lei received the Ph.D.degree in management science from Tamkang University,New Taipei City,Taiwan.She is currently working as a Researcher in Ditmanson Medical Foundation Chia-Yi Christian Hospital,Taiwan.Her research interests include management science,and process capability analysis.She has published research articles in refereed journals including International Journal of Advanced Manufacturing Technology,Applied Mathematics and Computation,Proceedings of the Insti-tution of Mechanical Engineers,Part B,Journal of Engineering Manufacture, Mathematical Problems in Engineering,and Applied Mathematical Modeling.。