應(yīng)用數(shù)學(xué)專業(yè)中英文翻譯

上海電力學(xué)院本科畢業(yè)設(shè)計（英文翻譯）英文原文：EstimationsForASimpleStep-StressModelWithProgressivelyType-IICensoredData院系：能源與環(huán)境工程學(xué)院專業(yè)年級：機(jī)械設(shè)計制造及其自動化2008級學(xué)生姓名：楊曉晨學(xué)號：200829722012年5InternationalJournalofReliability,QualityandSafetyEngineeringVol.12,No.5(2005)385–395?WorldScienti?cPublishingCompanyESTIMATIONSFORASIMPLESTEP-STRESSMODELWITHPROGRESSIVELYTYPE-IICENSOREDDATASHUO-JYEWU?andHSIU-MEILEEDepartmentofStatistics,TamkangTamsui,Taipei251Taiwan?shuo@.twDAR-HSINCHENGraduateInstituteNationalHsinchuCity300,TaiwanReceived1January2005Revised23May2005Withtoday’shightechnology,somelifetestsresultinnoorveryfewfailuresbytheendoftest.Insuchcases,anapproachistodolifetestathigher-than-usualstressconditionsinordertoobtainfailuresquickly.Thisstudydiscussesthepointandintervalestimationsofparametersonthesimplestep-stressmodelinacceleratedlifetestingwithprogressivetypeIIcensoring.Anexponentialfailuretimedistributionwithmeanlifethatisalog-linearfunctionofstressandacumulativeexposuremodelareconsidered.Wederivethemaximumlikelihoodestimatorsofthemodelparameters.Confidenceintervalsforthemodelparametersareestablishedbyusingpivotalquantityandcanbeappliedtoanysamplesize.Anumericalexampleisinvestigatedtoillustratetheproposedmethods.Keywords:Acceleratedlifetest;confidenceinterval;exponentialdistribution;maximumlikelihoodmethod;pivotalquantity;progressivetypeIIcensoring.IntroductionAcceleratedlifetest(ALT)isoftenusedforreliabilityanalysis.Testunitsarerunathigher-than-usualstressconditionsinordertoobtainfailuresquickly.Amodelrelatinglifelengthtostressisfittedtotheacceleratedfailuretimesandthenextrapolatedtoestimatethefailuretimedistributionunderusualconditions.ThestressloadinginanALTcanbeappliedvariousways.Theyincludeconstantstress,stepstress,andrandomstress.Nelson(Ref.10,Chap.1)discussedtheiradvantagesanddisadvantages.Instep-stressscheme,atestunitissubjectedtosuccessivelyhigherlevelsofstress.Atestunitstartsataspecifiedlowstressforaspecifiedlengthoftime.Ifitdoesnotfail,stressonitisraisedandheldaspecifiedtime.Thestressisthusincreasedstepbystepuntilthetestunitfails.Generallyalltestunitsgothroughthesamespecifiedpatternofstresslevelsandtesttimes.Thesimpleststep-stressALTusesonlytwostresslevelsandwecallsimplestep-stressALT.Thestatisticalinferenceinthissimplestep-stressALThasbeeninvestigatedbyseveralauthorssuchasTangetal.，11KhamisandHiggins，6Xiong，12YeoandTang，13Gouno，5McSorleyetal.，8DharmadhikariandRahman，4andAlhadeedandYang.1。InALT,testsareoftenstoppedbeforeallunitsfail.Theestimatefromthecensoreddataarelessaccuratethanthosefromcompletedata.However,thisismorethano?setbythereducedtesttimeandexpense.OneofthemostcommoncensoringschemesistypeIIcensoring.AtypeIIcensoredsamplehasobservedonlythem(1≤m≤n)smallestobservationsinarandomsampleofnunits.Ifanexperimenterdesirestoremoveliveunitsatpointsotherthanthe?nalterminationpointofthelifetest,theabovedescribedschemewillnotbeofusetotheexperimenter.TypeIIcensoringdoesnotallowforunitstoberemovedfromthetestatthepointsotherthanthe?nalterminationpoint.However,thisallowancewillbedesirable,asinthecaseofaccidentalbreakageoftestunits,inwhichthelossofunitsatpointsotherthantheterminationpointmaybeunavoidable.Intermediateremovalmayalsobedesirablewhenacompromiseissoughtbetweentimeconsumptionandtheobservationofsomeextremevalues.Thesereasonsleadusintotheareaofprogressivecensoring.Consideranexperimentinwhichnindependentunitsareplacedonatestattimezero,andthefailuretimesoftheseunitsarerecorded.Supposethatmfailuresaregoingtobeobserved.Whenthe?rstfailureisobserved,ofthesurvivingunitsarerandomlyselectedandremoved.Atthesecondobservedfailure,ofthesurvivingunitsarerandomlyselectedandremoved.Thisexperimentstopsatthetimewhenthem-thfailureisobservedandtheremaining…survivingunitsareallremoved.ThemorderedobservedfailuretimesarecalledprogressivelytypeIIcensoredorderstatisticsofsizemfromasampleofsizenwithcensoringscheme().Notethatif…,thenwhichcorrespondstothetypeIIcensoring,andif…,thenwhichcorrespondstothecompletesample.Inthisstudy,weconsiderpointandintervalestimationsforthesimplestepstressALTwith(1)progressivetypeIIcensoring,(2)anexponentialfailuretimedistributionataconstantstress,and(3)thecumulativeexposuremodel.InSec.2,wedescribethemodelandsomenecessaryassumptions.WeusethemaximumlikelihoodmethodtoobtainthepointestimatorsofthemodelparametersinSec.3.TheconfidenceintervalsforthemodelparametersarederivedinSec.4.AnumericaldatasetisstudiedtoillustratetheinferentialprocedureinSec.5.ModelandAssumptionsLetusconsiderthefollowingsimplestep-stressacceleratedlife-testingschemewithprogressivetypeIIcensoring:Supposenrandomlyselectedunitsaresimultaneouslyplacedonalifetestatstresssetting;thefailuretimesofthosethatfailinatimeintervalareobservedandsomesurvivingunitsareremovedwhenafailureoccurs;startingfromtime,thenon-removedsurvivingunitsareputtoadifferentstresssetting;thefailuretimesofthosethatfailareobservedandsomesurvivingunitsareremovedwhenafailureoccurs;atthetimeofthemthfailure,thelifetestisstopped.Foranystress,thefailuretimedistributionofthetestunitisanexponentialdistribution.Atstresslevel,themeanlifeofatestunitisalog-linearfunctionofstress.Thatis,，(1)Heretheandareunknownparameters.Thelog-linearfunctionisacommonchoiceforthelife-stressrelationshipsinceitincludesboththepower-lawandtheArrhenius-relationasspecialcases.Furthermore,failuresoccuraccordingtoacumulativeexposuremodel.Thatis,theremaininglifeofaunitdependsonlyontheexposureithasseen,andtheunitdoesnotrememberhowtheexposurewasaccumulated.(seeMillerandNelson9)Frompreviousassumptions,thecumulativedistributionfunctionofatestunitundersimplestep-stresstestis:whereandisthesolutionof.Hence,theprobabilitydensityfunctionofatestunitis(2)MaximumLikelihoodEstimationLet……beaprogressivelytypeIIcensoredsamplewithcensoringschemeromasimplestep-stressALT.Thatis,failuretimesofthetestunitsareobservedwhiletestingatstress,,andisthetotalnumberoffailures.Thus,thelikelihoodfunctionisgivenby,where……and.By(1),thelog-likelihoodfunctionforunknownparametersβ0andβ1maythenbewrittenas,for,,(3)where,and.letand.Wethenfindthatthemaximumlikelihoodestimators(MLEs)forandare,(4)and.(5)respectively.Undermildregularityconditions,anyofseveralmaximumlikelihoodlargesampleproceduresmightbeusedtomakeinferencesaboutand.Onepossibilityistoemploytheasymptoticnormalapproximationtoobtainconfidenceintervalsforand.WenowderivetheFisherinformationmatrix.From(3),wehave,(6),(7)and.(8)ToobtaintheFisherinformation,weneedtheexpectationsof(6),(7),and(8).Togetthese,letusfirstconsiderthefollowingtransformation.IfarandomvariableXhastheprobabilitydensityfunctionasin(2),thenXiong12showedthattherandomvariableisexponentiallydistributedwithmean1.Convertallintothrough(9).Then……isaprogressivelytypeIIcensoredsamplefromthestandardexponentialdistribution.FromBalakrishnanandAggarwala(Ref.2,p.19),wehavetheexpectationofiswhereThus,theFisherinformationiswhereand.TheFisherinformationcanbeinvertedtogettheasymptoticvariance-covariancematrixoftheMLEsaswhere，，and.ItfollowsformBickelandDoksum(Ref.3,p.398)thatisaconsistentestimatorof.Therefore,theapproximateconfidenceintervalsforandareand,respectively,whereistheupperpercentagepointofthestandardnormaldistribution.Remark1.Inpractice,canbe0or.ifandthelog-likelihoodfunctionbecomes.Hence,noMLEforandexists.TheMLEforis.Ifand,thelog-likelihoodfunctionis.Notethat,regardlessofthevaluesof,thelog-likelihoodfunctionisanincreasingfunctionof.Hence,noMLEforandexists.TheMLEforis.IntervalEstimationsInthissection,anexactconfidenceintervalforandanatleastconfidenceintervalforareconstructed.Considerthefollowingtransformation:(10)BalakrishnanandAggarwala(Ref.2,p.18)showedthatthegeneralizedspacings,asdefinedin(10),areindependentandidenticallydistributedasthestandardexponentialdistribution.Hence,bytheTheorem4.5.1inLawless,hasachi-squaredistributionwith2degreesoffreedomandhasachi-squaredistributionwith2m-2degreesoffreedom.WecanalsoshowthatUandVareindependent.Inthefollowingdiscussion,letbetheupperapercentagepointoftheFdistributionwithanddegreesoffreedomandletbetheupperapercentagepointofthechi-squaredistributionwithdegreesoffreedom.4.1.ConfidenceIntervalsforConsiderthecasethat.Let,where,and.Itiseasytoshowthatand,hencehasachi-squaredistributionwith2m-2degreesoffreedom.Itisalsoeasytoseethathasachi-squaredistributionwith2degreesoffreedom,andandareindependent.Now,wearegoingtoderiveaconfidenceintervalfor.Considerthepivotalquantity.Letand.For,wehaveHence,if,aconfidenceintervalforis,where.(11)If，aconfidenceintervalforis.NotethatthepreviousconfidenceintervalforisvalidundertheconditionHowever,inpractice,canbe0orm.Thus,wehavethefollowingtworemarks.Remark2.When,considerthepivotalquantity.where.Thentheconfidenceintervalforis,for，anemptysetelsewhere.Remark3.whenand,considerthepivotalquantity.Let.Then,theconfidenceintervalforis,for,elsewhere.4.2.ConfidenceIntervalforSupposethat.Wehaveisdistributedas,isdistributedas,andandareindependent.Let,,and.For,wehaveHence,if,anatleastconfidenceintervalforis,where,(12)and.(13)If,anatleastconfidenceintervalforis.Notethattheabovediscussionisunderthecondition.However,inpractice,canbe0orm.Therefore,wehavethefollowingtworemarks.Remark4.When,wehaveHence,anatleastconfidenceintervalforis,for,anemptysetelsewhere.Remark5.Whenand,wehaveThus,anatleastconfidenceintervalforis,for,where，and.If,theconfidenceintervalis.AnIllustrativeExampleToillustratetheuseofthemethodsgiveninthispaper,Table1presentsthesimulateddatafromasimplestep-stressALTmodelwithprogressivetypeIIcensoring.ThesedataweresimulatedbygeneratingaprogressivelytypeIIcensoredsamplefromthestandardexponentialdistributionusingthealgorithmpresentedinBalakrishnanandAggarwala(Ref.2,p.32),andthenthetransformation(9)isusedtogetthesamplefrommodel(2).Wechoose,,,,,,,andcensoringschemelistedinTable1.Table1.Simulatedfailuretimedata.stress123456781.431.502.746.507.6720020002stress91011121314151617181910.0110.0710.3010.3410.3510.5210.6210.6410.8210.8410.9604000020000stress202122232425262728293011.3911.6012.2912.3613.9614.2014.2915.0215.0516.8217.5530002000102Thenumbersoffailuresobservedatstressandatstressareand,respectively.Using(4)and(5),weareabletocalculatetheMLEsofand,andtherefore,theestimatesareandInadditiontopointestimation,wemayalsoconsiderintervalestimationsofand.A90%confidenceintervalforisobtainedasbyusing(11).Wealsoobtainfrom(12)and(13)thatanatleast90%confidenceintervalforis.AcknowledgementsTheauthorswouldliketothanktheEditorandrefereeforprovidinghelpfulcomments.TheworkofthefirstauthorwaspartiallysupportedbytheNationalScienceCouncilofROCgrantNSC89-2118-M-032-016.References1.A.A.AlhadeedandS.-S.Yang,Optimalsimplestep-stressplanforcumulativeexposuremodelusinglog-normaldistribution,IEEETransactionsonReliability54(2005)64–68.2.N.BalakrishnanandR.Aggarwala,ProgressiveCensoring—Theory,Methods,andApplications(Birkh¨auser,Boston,2000).3.P.J.BickelandK.A.Doksum,MathematicalStatistics:BasicIdeasandSelectedTopics,Vol.1,2ndedn.(PrenticeHall,UpperSaddleRiver,4.A.D.DharmadhikariandM.M.Rahman,Amodelforstep-stressacceleratedlifetesting,NavalResearchLogistics50(2003)841–868.5.E.Gouno,Aninferencemethodfortemperaturestep-stressacceleratedlifetesting,QualityandReliabilityEngineeringInternational17(2001)11–18.6.I.H.KhamisandJ.J.Higgins,Anewmodelforstep-stresstesting,IEEETransactionsonReliability47(1998)131–134.7.J.F.Lawless,StatisticalModelsandMethodsforLifetimeData,2ndedn.(JohnWiley&Sons,NewYork,2003).8.E.O.McSorley,J.-C.LuandC.-S.Li,Performanceofparameter-estimatesinstep-stressacceleratedlife-testswithvarioussample-sizes,IEEETransactionsonReliability51(2002)271–277.9.R.MillerandW.B.Nelson,Optimumsimplestepstressplansforacceleratedlifetesting,IEEETransactionsonReliability32(1983)59–65.10.W.Nelson,AcceleratedTesting:StatisticalModels,TestPlans,andDataAnalysis(JohnWiley&Sons,NewYork,1990).11.L.C.Tang,Y.S.Sun,T.N.GohandH.L.Ong,Analysisofstep-stressacceleratedlife-testdata:Anewapproach,IEEETransactionsonReliability45(1996)69–74.12.C.Xiong,Inferencesonasimplestep-stressmodelwithtype-IIcensoredexponentialdata,IEEETransactionsonReliability47(1998)142–146.13.K.-P.YeoandL.-C.Tang,Planningstep-stresslife-testwithatargetaccelerationfactor,IEEETransactionsonReliability48(1999)61–67.

AbouttheAuthorsShuo-JyeWuisaProfessorintheDepartmentofStatisticsatTamkangUniversity.HeobtainedhisPh.D.inStatisticsfromtheUniversityofWisconsin-Madison.HisprofessionalinterestsareinthedevelopmentandapplicationofstatisticalmethodologyHsiu-MeiLeeobtainedherPh.D.inManagementSciencesfromTamkangUniversityin1996.SheiscurrentlyemployedasanAssociateProfessorintheDepartmentofStatisticsatDar-HsinChenisanAssociateProfessorintheGraduateInstituteofFinanceatNationalChiaoTungInternationalJournalofReliability,QualityandSafetyEngineeringVol.12,No.5(2005)385–395?WorldScienti?cPublishingCompany逐次定數(shù)截尾數(shù)據(jù)下對一個簡單的步進(jìn)應(yīng)力模型的估計吳碩杰，李秀美淡江大學(xué)統(tǒng)計學(xué)系臺灣臺北淡水251號*shuo@.tw陳達(dá)新國立交通大學(xué)財務(wù)金融研究所臺灣新竹市300號2005年12005年5在今天的高新技術(shù)下，一些壽命試驗在測試結(jié)束后沒有或很少會產(chǎn)生失效。在這種情況下，為了獲得迅速地實現(xiàn)產(chǎn)品失效，一種方法是在高于正常應(yīng)力的條件下進(jìn)行壽命試驗。本研究目的在于探討在逐次定數(shù)截尾加速壽命試驗中的簡單步進(jìn)應(yīng)力模型的參數(shù)的點估計和區(qū)間估計。通過一個應(yīng)力的線性分布和一個累積失效模型下一個有效壽命的平均壽命分布，我們得出模型參數(shù)的極大似然估計量。模型參數(shù)的置信區(qū)間通過使用樞軸量而被建立，并且可以適用于任何大小的樣本。通過研究了一個數(shù)值例子來說明所提出的方法。關(guān)鍵詞：加速壽命試驗；置信區(qū)間；指數(shù)分布；極大似然估計法；樞軸量；逐次定數(shù)截尾。介紹加速壽命試驗(ALT)經(jīng)常被用于可靠性分析。為了使衰減更快速，試驗裝置運行在一個高于正常壓力的條件下。一個有關(guān)使用壽命的應(yīng)力模型被用來加速衰減時間，然后推算估計在正常條件下的失效時間分布。ALT可以應(yīng)用與多種方式應(yīng)力加載。它們包括恒定應(yīng)力，步進(jìn)應(yīng)力、和隨機(jī)應(yīng)力。Nelson(參考文獻(xiàn)10，第一章)討論了它們的優(yōu)點和缺點。在步進(jìn)應(yīng)力方案中，試驗裝置逐次在更高的應(yīng)力下進(jìn)行試驗。試驗裝置開始在一個特定的低應(yīng)力下進(jìn)行試驗，持續(xù)一定的時間。如果它不失效，加在裝置上的應(yīng)力升高并維持一定時間。應(yīng)力從而逐次增加，一直到試驗裝置的失效。一般所有的試驗裝置都經(jīng)歷同樣的指定模式的應(yīng)力水平和測試時間。最簡單的步進(jìn)應(yīng)力ALT僅僅使用兩個應(yīng)力水平，我們稱之為簡單步進(jìn)應(yīng)力ALT。對這個簡單步進(jìn)應(yīng)力ALT的統(tǒng)計推斷已被如下幾位作者調(diào)查，諸如Tang等，11Khamis和Higgins，6Xiong，12Yeo和Tang，13Gouno，5McSorleyetal.，8Dharmadhikari和Rahman，4和Alhadeed和Yang.1。在ALT中，試測試往往在所有裝置都已失效前就停止了。相比于完整數(shù)據(jù)，從截尾數(shù)據(jù)中得到的估計是不精確的。然而，這難以抵消減少的測試時間和費用。最常見的一種截尾方案是定數(shù)型截尾方案。一個定數(shù)型截尾樣品只觀測記錄隨機(jī)抽樣n個單位的m(1≤m≤n)個最小的觀察值。如果試驗者希望移除壽命試驗的最后終止點和其他的正在進(jìn)行試驗的裝置，上述計劃將不能再被試驗者使用。定數(shù)型截尾不允許從測試中刪除最后終止點以外的點單位。然而，這還算可取的，就像在意外破損情況下的試驗裝置，在其中損失的單位中除了終止點外可能都是不可避免的。在消耗的時間和觀察一些極端值之間尋求妥協(xié)時，中間去除，也可能是可取的。這些原因?qū)е铝宋覀冞M(jìn)入逐次截尾的領(lǐng)域?？紤]一個在時間零點的時候?qū)個獨立單位被放置在一個測試的試驗，且這些失效次數(shù)都被記錄。假設(shè)m的失效要觀察。當(dāng)觀察第一次失效時，的存活單位隨機(jī)選擇并刪除。在觀察第二個失效時，的存活單位隨機(jī)選擇并刪除。這個試驗在時間停止時觀察m次失效，其余…存活單位都刪除。在m次觀察失效時間，被稱為是從大小為n的樣本用截尾方案（）逐次定數(shù)截尾大小為m的有序統(tǒng)計量。注意，如果…，然后，是對應(yīng)定數(shù)型截尾，并且如果…，那么，就是對應(yīng)完整的樣本。在本研究中，我們認(rèn)為點估計和區(qū)間靠(1)逐次定數(shù)截尾，(2)正常盈利下恒定應(yīng)力指數(shù)的失效時間分布，以及(3)累積失效模型來估計簡單步進(jìn)應(yīng)力ALT。在第2節(jié)，我們描述了模型以及一些必要的假設(shè)。在第3節(jié)，我們使用極大似然法獲得了模型參數(shù)的點估計。在第4節(jié)，推導(dǎo)出了置信區(qū)間的模型參數(shù)。在第5節(jié)，對一個具體數(shù)值的數(shù)據(jù)集進(jìn)行研究，并說明推理過程。模型和假設(shè)讓我們考慮一下用逐次定數(shù)截尾方法實施以下簡單步進(jìn)應(yīng)力加速壽命試驗方案：假設(shè)隨機(jī)選擇的單位n在應(yīng)力設(shè)定下同時放到壽命試驗上；那些失效次數(shù)無法在時間區(qū)間內(nèi)觀察和當(dāng)發(fā)生失效時一些存活單位已經(jīng)被移除；從時間開始，那些沒被刪除的存活單位被放到一個不同的應(yīng)力設(shè)定上；那些失效次數(shù)被觀察和當(dāng)失效發(fā)生時一些存活單位被移除；在第m次失效時，壽命試驗將被停止。在任何應(yīng)力下，那些測試單位的失效壽命分布都是一個指數(shù)分布。在應(yīng)力水平下，平均壽命測試裝置是一種應(yīng)力的對數(shù)線性函數(shù)。那是，，(1)這里的和是未知參數(shù)。這個線性函數(shù)是一種常見的選擇應(yīng)力壽命關(guān)系，因為作為特殊情況它既包括冪規(guī)律和Arrhenius關(guān)系。此外，按照累積失效模型已經(jīng)發(fā)生失效。換言之，其裝置剩余的壽命的只取決于已發(fā)生的失效，并裝置不記得如何累積失效。(見MillerandNelson9)從先前的假設(shè)，測試裝置的累積分布函數(shù)在簡單步進(jìn)應(yīng)力壽命測試是:這里和是的解。因此，最合適測試裝置的概率密度函數(shù)是(2)極大似然估計設(shè)……是用截尾方案從一個簡單步進(jìn)應(yīng)力ALT中得到的逐次定數(shù)截尾樣本。也就是說，當(dāng)測試在應(yīng)力，觀察到測試裝置的失效時間，并且是失效的總數(shù)。因此，似然函數(shù)被如下給出：，這里……和。根據(jù)(1)，對于未知參數(shù)和的對數(shù)似然函數(shù)可以被寫成，對于時，，(3)其中，和。設(shè)和。然后我們會發(fā)現(xiàn)和的極大似然估計量（MLEs）分別是，(4)和。(5)在輕微規(guī)律條件下，任何幾個最大似然大樣本程序都可能會被用來推斷和。一個最大的可能就是采用逐步正態(tài)近似獲得和的置信區(qū)間?，F(xiàn)在我們得出了Fisher信息矩陣。從公式（3），有，(6)，(7)和。(8)為了獲得Fisher信息量，我們需要(6)(7)(8)的期望。為了得到這些，讓我們首先考慮以下的信息量。如果一個隨機(jī)變量在(2)中有概率密度函數(shù)，那么Xiong12表示這種隨機(jī)變量是平均為1的指數(shù)分布。通過(9)把所有的轉(zhuǎn)換成。然后……是從標(biāo)準(zhǔn)指數(shù)分布中得到的逐次定數(shù)截尾樣本。從Balakrishman和Aggarwala（參考文獻(xiàn)2，第19頁）中，我們可以得到的期望是這里。因此，F(xiàn)isher信息量是其中和。Fisher信息量能夠轉(zhuǎn)換成如下MLEs的漸進(jìn)方差-協(xié)方差矩陣其中，，和。它遵循了Bickel和Doksum（參考文獻(xiàn)3，第398頁）所說的，是的一個相合估計。因此，和的近似置信區(qū)間分別是和，其中是標(biāo)準(zhǔn)正態(tài)分布上的個百分點。備注1.在實際中，能夠是0或者。如果和，對數(shù)似然函數(shù)變?yōu)?。所以，沒有和存在的MLE。的MLE是。如果和，對數(shù)似然函數(shù)變?yōu)?。值得注意的是，無論的值為多少，對數(shù)似然函數(shù)是一個的增加函數(shù)。所以，沒有和存在的MLE。的MLE是。區(qū)間估計在這節(jié)中，一個確切的置信區(qū)間和至少的置信區(qū)間被構(gòu)造?？紤]以下的信息量：(10)BalakrishnanandAggarwala（參考文獻(xiàn)2，第18頁）顯示，廣義的間距，在(10)中定義，獨立同分布為標(biāo)準(zhǔn)的指數(shù)分布。所以，根據(jù)在Lawless11中4.5.1的定理，有2個自由度的卡方分布和有2m-2自由度的卡方分布。我們可以發(fā)現(xiàn)和是獨立的。在下面的討論，設(shè)在有和個自由度的分布的一個百分點上并且設(shè)在有個自由度的卡方分布的一個百分點上。4.1.的置信區(qū)間考慮的情況。設(shè)，其中，和。簡單表現(xiàn)了，并且，于是有一個具有2m-2自由度的卡方分布。顯而易見，也有一個具有2個自由度的卡方分布，并且和是獨立的。現(xiàn)在，我們將要得出的一個置信區(qū)間。考慮到樞軸量。設(shè)和。在，我們得到故，如果，一個的的置信區(qū)間是，其中。(11)如果，一個的置信區(qū)間是。注意，先前的置信區(qū)間在情況下是有效的。然而，在實際中，可能是0或者。因為，我們有了以下兩個備注。備注2.當(dāng)，考慮到樞軸量。其中。然后，的置信區(qū)間是，對于，空集在別處。備注3.當(dāng)和，考慮到樞軸量。設(shè)。然后，的置信區(qū)間是，對于，在別處。4.2.的置信區(qū)間假設(shè)。我們得到被分布成，被分布成，并且和是獨立的。設(shè)，，和。對于，我們得到故，如果，一個的至少置信區(qū)間是，其中，(12)和。(13)如果，一個的至少置信區(qū)間是。值得注意的是，以上討論是基于情況下的。然而，在實際中，可能是0或者。因為，我們有了以下兩個備注。備注4.當(dāng)，我們得到因此，一個的至少置信區(qū)間是，對于，空集在別處。備注5.當(dāng)和，我們得到由此，一個的至少置信區(qū)間是，對于，其中，和。如果，置信區(qū)間是。一個說明性的例子為了說明本文給出方法的使用，表1從一個虛擬的步進(jìn)ALT模型中用逐次定數(shù)截尾法列出了模擬數(shù)據(jù)。在Balakrishnan和Aggarwala（參考文獻(xiàn)2，第32頁）中，這些數(shù)據(jù)通過標(biāo)準(zhǔn)正態(tài)分布算法，被模擬生成逐次定數(shù)截尾樣本，然后從模型(2)中得到樣本的信息量。我們選擇，，，，，，，并且截尾方案被列于表1。表1模擬失效時間數(shù)據(jù)應(yīng)力123456781.431.502.746.507.6720020002應(yīng)力91011121314151617181910.0110.0710.3010.3410.3510.5210.6210.6410.8210.8410.9604000020000應(yīng)力202122232425262728293011.3911.6012.2912.3613.9614.2014.2915.0215.0516.8217.5530002000102在應(yīng)力和下觀察到的失效分別是和。用(4)和(5)，我們能夠計算出和的MLEs，因此，估計得到和除去點估計，我們也能夠得出和的置信區(qū)間。的一個90%置信區(qū)間被獲得如下通過采用式(11)，我們也可以從(12)(13)獲得的至少有90%