(cid:2) AcceleratedLifeTestingofOne-shotDevices (cid:2) (cid:2) (cid:2) (cid:2) Accelerated Life Testing of One-shot Devices Data Collection And Analysis Narayanaswamy Balakrishnan McMasterUniversity Hamilton,Canada (cid:2) (cid:2) Man Ho Ling TheEducationUniversityofHongKong TaiPo,HongKongSAR,China Hon Yiu So UniversityofWaterloo Waterloo,Canada (cid:2) (cid:2) Thisfirsteditionfirstpublished2021 ©2021byJohnWileyandSons,Inc. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingor otherwise,exceptaspermittedbylaw.Adviceonhowtoobtainpermissiontoreusematerialfromthis titleisavailableathttp://www.wiley.com/go/permissions. Therightof NarayanaswamyBalakrishnan,ManHoLing,andHonYiuSotobeidentifiedasthe author(s)ofthisworkhasbeenassertedinaccordancewithlaw. 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LibraryofCongressCataloging-in-PublicationData Names:Balakrishnan,Narayanaswamy.,1956-author.|Ling,ManHo,author.|So,Hon Yiu,author. Title:Acceleratedlifetestingofone-shotdevices:datacollectionand analysis/NarayanaswamyBalakrishnan,McMasterUniversity,Hamilton, Canada,ManHoLing,TheEducationUniversityofHong,Kong,New Territories,HongKong,HonYiuSo,UniversityofWaterloo,Waterloo, Canada. Description:Firstedition.|Hoboken,NJ,USA:Wiley,2021.|Includes bibliographicalreferencesandindex. Identifiers:LCCN2020035725(print)|LCCN2020035726(ebook)|ISBN 9781119664000(cloth)|ISBN9781119664017(adobepdf)|ISBN 9781119663942(epub) Subjects:LCSH:Acceleratedlifetesting.|Failureanalysis(Engineering) Classification:LCCTA169.3.B352021(print)|LCCTA169.3(ebook)|DDC 620/.00452–dc23 LCrecordavailableathttps://lccn.loc.gov/2020035725 LCebookrecordavailableathttps://lccn.loc.gov/2020035726 CoverDesign:Wiley CoverImage:©Piergiov/GettyImages Setin9.5/12.5ptSTIXTwoTextbySPiGlobal,Chennai,India 10 9 8 7 6 5 4 3 2 1 (cid:2) (cid:2) Withgreatloveandaffection,wededicatethisbookto SarahandJuliaBalakrishnan,andColleenCutler NB GraceChu,SophiaLing,andSheldonLing MHL TianFengandVictoriaSo HYS (cid:2) (cid:2) (cid:2) (cid:2) vii Contents Preface xi AbouttheCompanionWebsite xiii 1 One-ShotDeviceTestingData 1 1.1 BriefOverview 1 1.2 One-ShotDevices 1 1.3 AcceleratedLife-Tests 3 1.4 ExamplesinReliabilityandSurvivalStudies 4 (cid:2) (cid:2) 1.4.1 Electro-ExplosiveDevicesData 4 1.4.2 GlassCapacitorsData 5 1.4.3 SolderJointsData 5 1.4.4 Grease-BasedMagnetorheologicalFluidsData 6 1.4.5 MiceTumorToxicologicalData 7 1.4.6 ED01ExperimentData 7 1.4.7 SerialSacrificeData 7 1.5 RecentDevelopmentsinOne-ShotDeviceTestingAnalysis 10 2 LikelihoodInference 13 2.1 BriefOverview 13 2.2 UnderCSALTsandDifferentLifetimeDistributions 13 2.3 EM-Algorithm 14 2.3.1 ExponentialDistribution 16 2.3.2 GammaDistribution 18 2.3.3 WeibullDistribution 21 2.4 IntervalEstimation 26 2.4.1 AsymptoticConfidenceIntervals 26 2.4.2 ApproximateConfidenceIntervals 28 2.5 SimulationStudies 30 2.6 CaseStudieswithRCodes 41 (cid:2) (cid:2) viii Contents 3 BayesianInference 47 3.1 BriefOverview 47 3.2 BayesianFramework 47 3.3 ChoiceofPriors 49 3.3.1 LaplacePrior 49 3.3.2 NormalPrior 49 3.3.3 BetaPrior 50 3.4 SimulationStudies 51 3.5 CaseStudywithRCodes 59 4 ModelMis-SpecificationAnalysisandModelSelection 65 4.1 BriefOverview 65 4.2 ModelMis-SpecificationAnalysis 65 4.3 ModelSelection 66 4.3.1 AkaikeInformationCriterion 66 4.3.2 BayesianInformationCriterion 67 4.3.3 Distance-BasedTestStatistic 68 4.3.4 ParametricBootstrapProcedureforTestingGoodness-of-Fit 70 4.4 SimulationStudies 70 4.5 CaseStudywithRCodes 76 (cid:2) (cid:2) 5 RobustInference 79 5.1 BriefOverview 79 5.2 WeightedMinimumDensityPowerDivergenceEstimators 79 5.3 AsymptoticDistributions 81 5.4 RobustWald-typeTests 82 5.5 InfluenceFunction 83 5.6 SimulationStudies 85 5.7 CaseStudywithRCodes 91 6 Semi-ParametricModelsandInference 95 6.1 BriefOverview 95 6.2 ProportionalHazardsModels 95 6.3 LikelihoodInference 97 6.4 TestofProportionalHazardRates 99 6.5 SimulationStudies 100 6.6 CaseStudieswithRCodes 102 7 OptimalDesignofTests 105 7.1 BriefOverview 105 7.2 OptimalDesignofCSALTs 105 (cid:2) (cid:2) Contents ix 7.3 OptimalDesignwithBudgetConstraints 106 7.3.1 SubjecttoSpecifiedBudgetandTerminationTime 107 7.3.2 SubjecttoStandardDeviationandTerminationTime 107 7.4 CaseStudieswithRCodes 108 7.5 SensitivityofOptimalDesigns 113 8 DesignofSimpleStep-StressAcceleratedLife-Tests 119 8.1 BriefOverview 119 8.2 One-ShotDeviceTestingDataUnderSimpleSSALTs 119 8.3 AsymptoticVariance 121 8.3.1 ExponentialDistribution 121 8.3.2 WeibullDistribution 122 8.3.3 WithaKnownShapeParameter𝑤 124 2 8.3.4 WithaKnownParameterAboutStressLevel𝑤 125 1 8.4 OptimalDesignofSimpleSSALT 126 8.5 CaseStudieswithRCodes 128 8.5.1 SSALTforExponentialDistribution 128 8.5.2 SSALTforWeibullDistribution 131 9 Competing-RisksModels 141 (cid:2) (cid:2) 9.1 BriefOverview 141 9.2 One-ShotDeviceTestingDatawithCompetingRisks 141 9.3 LikelihoodEstimationforExponentialDistribution 143 9.3.1 WithoutMaskedFailureModes 144 9.3.2 WithMaskedFailureModes 147 9.4 LikelihoodEstimationforWeibullDistribution 149 9.5 BayesianEstimation 155 9.5.1 WithoutMaskedFailureModes 155 9.5.2 LaplacePrior 156 9.5.3 NormalPrior 157 9.5.4 DirichletPrior 157 9.5.5 WithMaskedFailureModes 158 9.6 SimulationStudies 159 9.7 CaseStudywithRCodes 165 10 One-ShotDeviceswithDependentComponents 173 10.1 BriefOverview 173 10.2 TestDatawithDependentComponents 173 10.3 CopulaModels 174 10.3.1 FamilyofArchimedeanCopulas 175 10.3.2 Gumbel–HougaardCopula 176 (cid:2) (cid:2) x Contents 10.3.3 FrankCopula 177 10.4 EstimationofDependence 180 10.5 SimulationStudies 181 10.6 CaseStudywithRCodes 184 11 ConclusionsandFutureDirections 187 11.1 BriefOverview 187 11.2 ConcludingRemarks 187 11.2.1 LargeSampleSizesforFlexibleModels 187 11.2.2 AccurateEstimation 188 11.2.3 GoodDesignsBeforeDataAnalysis 188 11.3 FutureDirections 189 11.3.1 WeibullLifetimeDistributionwithThresholdParameter 189 11.3.2 FrailtyModels 189 11.3.3 OptimalDesignofSSALTswithMultipleStressLevels 189 11.3.4 ComparisonofCSALTsandSSALTs 190 AppendixA DerivationofH(a,b) 191 i AppendixB ObservedInformationMatrix 193 (cid:2) (cid:2) AppendixC Non-IdentifiableParametersforSSALTsUnderWeibull Distribution 197 AppendixD OptimalDesignUnderWeibullDistributionswithFixed 𝒘 199 1 AppendixE ConditionalExpectationsforCompetingRisksModelUnder ExponentialDistribution 201 AppendixF Kendall’sTauforFrankCopula 205 Bibliography 207 AuthorIndex 217 SubjectIndex 221 (cid:2) (cid:2) xi Preface Lifetimeinformationobtainedfromone-shotdevicesisverylimitedastheentire data are either left- or right-censored. For this reason, the analysis of one-shot devicetestingdataposesaspecialchallenge.Thisbookprovidesseveralstatisti- calinferentialmethodsforanalyzingone-shotdevicelifetimedataobtainedfrom acceleratedlife-testsandalsodevelopsoptimaldesignsfortwomainstreamaccel- erated life-tests – constant-stress and step-stress accelerated life-tests – that are commonlyusedinreliabilitypractice.Thediscussionsprovidedinthebookwould (cid:2) enablereliabilitypractitionerstobetterdesigntheirexperimentsfordatacollec- (cid:2) tionfromefficientacceleratedlife-testswhentherearebudgetconstraintsinplace. Thisisimportantfromestimationandpredictionpointofviewassuchoptimal designswouldresultinasaccurateaninferenceaspossibleundertheconstraints imposed on the reliability experiment. Moreover, R codes are presented within eachchaptersothatuserscantryoutperformingtheirownanalysisonone-shot devicetestingdata. In addition, the inferential methods and the procedures for planning accel- erated life-tests discussed in this book are not only limited to one-shot devices alone but also can be extended naturally to accelerated life-tests with periodic inspections (interval-censoring) and those with continuous monitoring and censoring (right-censoring). The book finally concludes by highlighting some important issues and problems that are worth considering for further research. Thismaybeespeciallyusefulforresearchscholarsandnewresearchersinterested intakingonthisinterestingandchallengingareaofresearchinreliabilitytheory andpractice. Itispossiblethatsomepertinentresultsorreferencesgotomittedinthisbook, and we assure you that it is due to inadvertency on our part and not due to scientific antipathy. We will appreciate greatly if the readers inform us of any corrections/omissions, or any comments pertinent to any of the discussions in thebook! (cid:2) (cid:2) xii Preface OursincerethanksgototheentireWileyteam,Ms.MindyOkura-Marszycki, Ms.KathleenSantoloci,andMr.BrettKurzman,fortakinggreatinterestinthis project from day one, for all their help and encouragement during the whole course,andfortheirfineassistanceduringthefinalproductionstageofthebook. Ourthanksalsogotoourresearchcollaboratorsandgraduatestudentsfortheir incisive comments and queries, which always benefited us greatly and helped clarifysomeofourownideas!WeexpressoursincereappreciationtoMs.Elena Maria Castilla Gonzalez, a doctoral student of Professor Leandro Pardo in the Department of Statistics and Operations Research at Complutense University ofMadrid,Spain,forhercarefulreadingofChapter5andalsoforsharingwith ussomeRcodesthatshehaddevelopedconcerningrobustinferentialmethods forone-shotdevicetestanalyses.Lastbutnotleast,ourspecialthanksgotoour familiesfortheirpatienceandunderstanding,andforprovidingconstantsupport andencouragementduringourworkonthisbook! Finally, the first author (NB) wishes to state to his older daughter, Ms. Sarah Balakrishnan, that though she lost out on getting his Volvo car due to a major caraccident,sheshouldbeheartenedbythefactthattheaccidentresultedinthe germinationofhisinterestandideasonone-shotdevices(airbags),andultimately thisbooksolelydedicatedtothetopic! (cid:2) (cid:2) July,2020 NarayanaswamyBalakrishnan ManHoLing HonYiuSo (cid:2)