WeusethemaximumlikelihoodmethodtoobtainthepointestimatorsofthemodelparametersinSec..TheconfidenceintervalsforthemodelparametersarederivedinSec..AnumericaldatasetisstudiedtoillustratetheinferentialprocedureinSec...ModelandAssumptionsLetusconsiderthefollowingsimplestep-stressacceleratedlife-testingschemewithprogressivetypeIIcensoring:SupposenrandomlyselectedunitsaresimultaneouslyplacedonalifetestatseninvestigatedbyseveralauthorssuchasTangetal.,KhamisandHiggins,Xiong,YeoandTang,Gouno,McSorleyetal.,DharmadhikariandRahman,andAlhadeedandYang.。InALT,testsareoftenstoppedbeforeallunitsfail.Theestimatefromthecensoreddataarelessaccuratethanthosefromcompletedata.However,thisismorethanoffsetbythereducedtesttimeandexpense.OneofthemostcommoncensoringschemesistypeIIcensoring.AtypeIIcensoredsamplehasobservedonlythem(≤m≤n)smallestobservationsinarandomsampleofnunits.Ifanexperimenterdesirestoremoveliveunitsatpointsotherthanthefinalterminationpointofthelifetest,theabovedescribedschemewillnotbeofusetotheexperimenter.TypeIIcensoringdoesnotallowforunitstoberemovedfromthetestatthepointsotherthanthefinalterminationpoint.However,thisallowancewillbedesirable,asinthecaseofaccidentalbreakageoftestunits,inwhichthelossofunitsatpointsotherthantheterminationpointmaybeunavoidable.Intermediateremovalmayalsobedesirablewhenacompromiseissoughtbetweentimeconsumptionandtheobservationofsomeextremevalues.Thesereasonsleadusintotheareaofprogressivecensoring.Consideranexperimentinwhichnindependentunitsareplacedonatestattimezero,andthefailuretimesoftheseunitsarerecorded.Supposethatmfailuresaregoingtobeobserved.Whenthefirstfailureisobserved,rofthesurvivingunitsarerandomlyselectedandremoved.Atthesecondobservedfailure,rofthesurvivingunitsarerandomlyselectedandremoved.Thisexperimentstopsatthetimewhenthem-thfailureisobservedandtheremainingmrnrr…mrmsurvivingunitsareallremoved.ThemorderedobservedfailuretimesarecalledprogressivelytypeIIcensoredorderstatisticsofsizemfromasampleofsizenwithcensoringscheme(,...,mrr).Notethatifrr…mr,thenmrnmwhichcorrespondstothetypeIIcensoring,andifrr…mr,thennmwhichcorrespondstothecompletesample.Inthisstudy,weconsiderpointandintervalestimationsforthesimplestepstressALTwith()progressivetypeIIcensoring,()anexponentialfailuretimedistributionataconstantstress,and()thecumulativeexposuremodel.InSec.,wedescribethemodelandsomenecessaryassumptions.WeusethemaximumlikelihoodmethodtoobtainthepointestimatorsofthemodelparametersinSec..TheconfidenceintervalsforthemodelparametersarederivedinSec..AnumericaldatasetisstudiedtoillustratetheinferentialprocedureinSec...ModelandAssumptionsLetusconsiderthefollowingsimplestep-stressacceleratedlife-testingschemewithprogressivetypeIIcensoring:Supposenrandomlyselectedunitsaresimultaneouslyplacedonalifetestats且截尾方案,,...,rrr被列于表。表模拟失效时间数据应力viix........ir应力viix...........ir应力viix...........ir在应力v和v下观察到失效分别是n和n。用()和(),我们能够计算出和MLEs,因此,估计得到^.和^.除去点估计,我们也能够得出和置信区间。一个置信区间被获得如下.,.通过采用式(),我们也可以从()()获得至少有置信区间是.,.。致谢笔者想要感谢编辑和审稿人提供了高贵意见。第一作者工作部分得到了中华民国国家科学委员会资助,项目基金号NSC--M--。参考文献.A.A.AlhadeedandS.-S.Yang,Optimalsimplestep-stressplanforcumulativeexposuremodelusinglog-normaldistribution,IEEETransactionsonReliability()–..N.BalakrishnanandR.Aggarwala,ProgressiveCensoring—Theory,Methods,andApplications(Birkhauser,Boston,)..P.J.BickelandK.A.Doksum,MathematicalStatistics:BasicIdeasandSelectedTopics,Vol.,ndedn.(PrenticeHall,UpperSaddleRiver,NJ,)..A.D.DharmadhikariandM.M.Rahman,Amodelforstep-stressacceleratedlifetesting,NavalResearchLogistics()–..E.Gouno,Aninferencemethodfortemperaturestep-stressacceleratedlifetesting,QualityandReliabilityEngineeringInternational()–..I.H.KhamisandJ.J.Higgins,Anewmodelforstep-stresstesting,IEEETransactionsonReliability()–..J.F.Lawless,StatisticalModelsandMethodsforLifetimeData,ndedn.(JohnWiley&Sons,NewYork,)..E.O.McSorley,J.-C.LuandC.-S.Li,Performanceofparameter-estimatesinstep-stressacceleratedlife-testswithvarioussample-sizes,IEEETransactionsonReliability()–..R.MillerandW.B.Nelson,Optimumsimplestepstressplansforacceleratedlifetesting,IEEETransactionsonReliability()–..W.Nelson,AcceleratedTesting:StatisticalModels,TestPlans,andDataAnalysis(JohnWiley&Sons,NewYork,)..L.C.Tang,Y.S.Sun,T.N.GohandH.L.Ong,Analysisofstep-stressacceleratedlife-testdata:Anewapproach,IEEETransactionsonReliability()–..C.Xiong,Inferencesonasimplestep-stressmodelwithtype-IIcensoredexponentialdata,IEEETransactionsonReliability()–..K.-P.YeoandL.-C.Tang,Planningstep-stresslife-testwithatargetaccelerationfactor,IEEETransactionsonReliability()–.关于作者吴硕杰是在淡江大学统计系教授。在Wisconsin-Madison大学他获得了他统计学博士学位。他专业兴趣在统计方法可靠性问题发展和应用。李秀美年在淡江大学获得了管理科学博士学位。她目前受聘为淡江大学统计系副教授。她研究兴趣是应用统计。陈达新是国立交通大学财务金融研究所副教授。在Mississippi大学他获得了他财务博士学位。他专业兴趣在模拟金融资产波动中运用统计和计量模型。eninvestigatedbyseveralauthorssuchasTangetal.,KhamisandHiggins,Xiong,YeoandTang,Gouno,McSorleyetal.,DharmadhikariandRahman,andAlhadeedandYang.。InALT,testsareoftenstoppedbeforeallunitsfail.Theestimatefromthecensoreddataarelessaccuratethanthosefromcompletedata.However,thisismorethanoffsetbythereducedtesttimeandexpense.OneofthemostcommoncensoringschemesistypeIIcensoring.AtypeIIcensoredsamplehasobserve本科毕业设计(英文翻译)英文原文:EstimationsForASimpleStep-StressModelWithProgressivelyType-IICensoredData院系:能源与环境工程学院专业年级:机械设计制造及其自动化级学生姓名:学号:年月日InternationalJournalofReliability,QualityandSafetyEngineeringVol.,No.()–WorldScientificPublishingCompanyESTIMATIONSFORASIMPLESTEP-STRESSMODELWITHPROGRESSIVELYTYPE-IICENSOREDDATASHUO-JYEWU∗andHSIU-MEILEEDepartmentofStatistics,TamkangUniversityTamsui,TaipeiTaiwan∗shuo@stat.tku.edu.twDAR-HSINCHENGraduateInstituteofFinanceNationalChiaoTungUniversityHsinchuCity,TaiwanReceivedJanuaryRevisedMayWithtoday’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.,Chap.)discussedtheiradvantagesanddisadvantages.Instep-stressscheme,atestunitissubjectedtosuccessivelyhigherlevelsofstress.Atestunitstartsataspecifiedlowstressforaspecifiedlengthoftime.Ifitdoesnotfail,stressonitisraisedandheldaspecifiedtime.Thestressisthusincreasedstepbystepuntilthetestunitfails.Generallyalltestunitsgothroughthesamespecifiedpatternofstresslevelsandtesttimes.Thesimpleststep-stressALTusesonlytwostresslevelsandwecallsimplestep-stressALT.Thestatisticalinferenceinthissimplestep-stressALThasbeeninvestigatedbyseveralauthorssuchasTangetal.,KhamisandHiggins,Xiong,YeoandTang,Gouno,McSorleyetal.,DharmadhikariandRahman,andAlhadeedandYang.。InALT,testsareoftenstoppedbeforeallunitsfail.Theestimatefromthecensoreddataarelessaccuratethanthosefromcompletedata.However,thisismorethanoffsetbythereducedtesttimeandexpense.OneofthemostcommoncensoringschemesistypeIIcensoring.AtypeIIcensoredsamplehasobservedonlythem(≤m≤n)smallestobservationsinarandomsampleofnunits.Ifanexperimenterdesirestoremoveliveunitsatpointsotherthanthefinalterminationpointofthelifetest,theabovedescribedschemewillnotbeofusetotheexperimenter.TypeIIcensoringdoesnotallowforunitstoberemovedfromthetestatthepointsotherthanthefinalterminationpoint.However,thisallowancewillbedesirable,asinthecaseofaccidentalbreakageoftestunits,inwhichthelossofunitsatpointsotherthantheterminationpointmaybeunavoidable.Intermediateremovalmayalsobedesirablewhenacompromiseissoughtbetweentimeconsumptionandtheobservationofsomeextremevalues.Thesereasonsleadusintotheareaofprogressivecensoring.Consideranexperimentinwhichnindependentunitsareplacedonatestattimezero,andthefailuretimesoftheseunitsarerecorded.Supposethatmfailuresaregoingtobeobserved.Whenthefirstfailureisobserved,rofthesurvivingunitsarerandomlyselectedandremoved.Atthesecondobservedfailure,rofthesurvivingunitsarerandomlyselectedandremoved.Thisexperimentstopsatthetimewhenthem-thfailureisobservedandtheremainingmrnrr…mrmsurvivingunitsareallremoved.ThemorderedobservedfailuretimesarecalledprogressivelytypeIIcensoredorderstatisticsofsizemfromasampleofsizenwithcensoringscheme(,...,mrr).Notethatifrr…mr,then 本科毕业设计(英文翻译)英文原文:EstimationsForASimpleStep-StressModelWithProgressivelyType-IICensoredData院系:能源与环境工程学院专业年级:机械设计制造及其自动化2008级学生姓名:学号:2012年5月12日1InternationalJournalofReliability,QualityandSafetyEngineeringVol.12,No.5(2005)385–395©WorldScientificPublishingCompanyESTIMATIONSFORASIMPLESTEP-STRESSMODELWITHPROGRESSIVELYTYPE-IICENSOREDDATASHUO-JYEWU∗andH