Narayanaswamy Balakrishnan Visit to download the full and correct content document: https://ebookmass.com/product/accelerated-life-testing-of-one-shot-devices-data-coll ection-and-analysis-narayanaswamy-balakrishnan/
More products digital (pdf, epub, mobi) instant download maybe you interests ...
Chi-Squared Data Analysis and Model Testing for Beginners 1st Edition Carey Witkov
https://ebookmass.com/product/chi-squared-data-analysis-andmodel-testing-for-beginners-1st-edition-carey-witkov/
Residual Current Devices: Selection, Operation, and Testing Stanislaw Czapp
https://ebookmass.com/product/residual-current-devices-selectionoperation-and-testing-stanislaw-czapp/
Data Parallel C++ : Programming Accelerated Systems
Using C++ and SYCL James Reinders
https://ebookmass.com/product/data-parallel-c-programmingaccelerated-systems-using-c-and-sycl-james-reinders/
One Last Shot (Alaska Air One Rescue Book 1) Susan May
Warren
https://ebookmass.com/product/one-last-shot-alaska-air-onerescue-book-1-susan-may-warren/
Data Analysis for
the Life Sciences with R 1st Edition https://ebookmass.com/product/data-analysis-for-the-lifesciences-with-r-1st-edition/
Data Parallel C++: Programming Accelerated Systems Using C++ and SYCL 2nd Edition James Reinders
https://ebookmass.com/product/data-parallel-c-programmingaccelerated-systems-using-c-and-sycl-2nd-edition-james-reinders/
Nanostructured Multiferroics Raneesh Balakrishnan https://ebookmass.com/product/nanostructured-multiferroicsraneesh-balakrishnan/
Structural Adhesive Joints: Design, Analysis, and Testing K. L. Mittal
https://ebookmass.com/product/structural-adhesive-joints-designanalysis-and-testing-k-l-mittal/
Statistics for Ecologists Using R and Excel: Data Collection, Exploration,
https://ebookmass.com/product/statistics-for-ecologists-using-rand-excel-data-collection-exploration/
AcceleratedLifeTestingofOne-shotDevices AcceleratedLifeTestingofOne-shotDevices DataCollectionAndAnalysis NarayanaswamyBalakrishnan
McMasterUniversity Hamilton,Canada
ManHoLing
TheEducationUniversityofHongKong
TaiPo,HongKongSAR,China
HonYiuSo UniversityofWaterloo Waterloo,Canada
Thisfirsteditionfirstpublished2021 ©2021byJohnWileyandSons,Inc.
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingor otherwise,exceptaspermittedbylaw.Adviceonhowtoobtainpermissiontoreusematerialfromthis titleisavailableathttp://www.wiley.com/go/permissions.
TherightofNarayanaswamyBalakrishnan,ManHoLing,andHonYiuSotobeidentifiedasthe author(s)ofthisworkhasbeenassertedinaccordancewithlaw.
RegisteredOffice
JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,USA
EditorialOffice 111RiverStreet,Hoboken,NJ07030,USA
Fordetailsofourglobaleditorialoffices,customerservices,andmoreinformationaboutWiley productsvisitusatwww.wiley.com.
Wileyalsopublishesitsbooksinavarietyofelectronicformatsandbyprint-on-demand.Somecontent thatappearsinstandardprintversionsofthisbookmaynotbeavailableinotherformats.
LimitofLiability/DisclaimerofWarranty
Whilethepublisherandauthorshaveusedtheirbesteffortsinpreparingthiswork,theymakeno representationsorwarrantieswithrespecttotheaccuracyorcompletenessofthecontentsofthiswork andspecificallydisclaimallwarranties,includingwithoutlimitationanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales representatives,writtensalesmaterialsorpromotionalstatementsforthiswork.Thefactthatan organization,website,orproductisreferredtointhisworkasacitationand/orpotentialsourceof furtherinformationdoesnotmeanthatthepublisherandauthorsendorsetheinformationorservices theorganization,website,orproductmayprovideorrecommendationsitmaymake.Thisworkissold withtheunderstandingthatthepublisherisnotengagedinrenderingprofessionalservices.The adviceandstrategiescontainedhereinmaynotbesuitableforyoursituation.Youshouldconsultwith aspecialistwhereappropriate.Further,readersshouldbeawarethatwebsiteslistedinthisworkmay havechangedordisappearedbetweenwhenthisworkwaswrittenandwhenitisread.Neitherthe publishernorauthorsshallbeliableforanylossofprofitoranyothercommercialdamages,including butnotlimitedtospecial,incidental,consequential,orotherdamages.
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
10987654321
Withgreatloveandaffection,wededicatethisbookto
SarahandJuliaBalakrishnan,andColleenCutlerNB
GraceChu,SophiaLing,andSheldonLingMHL
TianFengandVictoriaSoHYS
Contents
Preface xi
AbouttheCompanionWebsite xiii
1One-ShotDeviceTestingData 1
1.1BriefOverview 1
1.2One-ShotDevices 1
1.3AcceleratedLife-Tests 3
1.4ExamplesinReliabilityandSurvivalStudies 4
1.4.1Electro-ExplosiveDevicesData 4
1.4.2GlassCapacitorsData 5
1.4.3SolderJointsData 5
1.4.4Grease-BasedMagnetorheologicalFluidsData 6
1.4.5MiceTumorToxicologicalData 7
1.4.6ED01ExperimentData 7
1.4.7SerialSacrificeData 7
1.5RecentDevelopmentsinOne-ShotDeviceTestingAnalysis 10
2LikelihoodInference 13
2.1BriefOverview 13
2.2UnderCSALTsandDifferentLifetimeDistributions 13
2.3EM-Algorithm 14
2.3.1ExponentialDistribution 16
2.3.2GammaDistribution 18
2.3.3WeibullDistribution 21
2.4IntervalEstimation 26
2.4.1AsymptoticConfidenceIntervals 26
2.4.2ApproximateConfidenceIntervals 28
2.5SimulationStudies 30
2.6CaseStudieswithRCodes 41
3BayesianInference 47
3.1BriefOverview 47
3.2BayesianFramework 47
3.3ChoiceofPriors 49
3.3.1LaplacePrior 49
3.3.2NormalPrior 49
3.3.3BetaPrior 50
3.4SimulationStudies 51
3.5CaseStudywithRCodes 59
4ModelMis-SpecificationAnalysisandModelSelection 65
4.1BriefOverview 65
4.2ModelMis-SpecificationAnalysis 65
4.3ModelSelection 66
4.3.1AkaikeInformationCriterion 66
4.3.2BayesianInformationCriterion 67
4.3.3Distance-BasedTestStatistic 68
4.3.4ParametricBootstrapProcedureforTestingGoodness-of-Fit 70
4.4SimulationStudies 70
4.5CaseStudywithRCodes 76
5RobustInference 79
5.1BriefOverview 79
5.2WeightedMinimumDensityPowerDivergenceEstimators 79
5.3AsymptoticDistributions 81
5.4RobustWald-typeTests 82
5.5InfluenceFunction 83
5.6SimulationStudies 85
5.7CaseStudywithRCodes 91
6Semi-ParametricModelsandInference 95
6.1BriefOverview 95
6.2ProportionalHazardsModels 95
6.3LikelihoodInference 97
6.4TestofProportionalHazardRates 99
6.5SimulationStudies 100
6.6CaseStudieswithRCodes 102
7OptimalDesignofTests 105
7.1BriefOverview 105
7.2OptimalDesignofCSALTs 105
7.3OptimalDesignwithBudgetConstraints 106
7.3.1SubjecttoSpecifiedBudgetandTerminationTime 107
7.3.2SubjecttoStandardDeviationandTerminationTime 107
7.4CaseStudieswithRCodes 108
7.5SensitivityofOptimalDesigns 113
8DesignofSimpleStep-StressAcceleratedLife-Tests 119
8.1BriefOverview 119
8.2One-ShotDeviceTestingDataUnderSimpleSSALTs 119
8.3AsymptoticVariance 121
8.3.1ExponentialDistribution 121
8.3.2WeibullDistribution 122
8.3.3WithaKnownShapeParameter ��2 124
8.3.4WithaKnownParameterAboutStressLevel ��1 125
8.4OptimalDesignofSimpleSSALT 126
8.5CaseStudieswithRCodes 128
8.5.1SSALTforExponentialDistribution 128
8.5.2SSALTforWeibullDistribution 131
9Competing-RisksModels 141
9.1BriefOverview 141
9.2One-ShotDeviceTestingDatawithCompetingRisks 141
9.3LikelihoodEstimationforExponentialDistribution 143
9.3.1WithoutMaskedFailureModes 144
9.3.2WithMaskedFailureModes 147
9.4LikelihoodEstimationforWeibullDistribution 149
9.5BayesianEstimation 155
9.5.1WithoutMaskedFailureModes 155
9.5.2LaplacePrior 156
9.5.3NormalPrior 157
9.5.4DirichletPrior 157
9.5.5WithMaskedFailureModes 158
9.6SimulationStudies 159
9.7CaseStudywithRCodes 165
10One-ShotDeviceswithDependentComponents 173
10.1BriefOverview 173
10.2TestDatawithDependentComponents 173
10.3CopulaModels 174
10.3.1FamilyofArchimedeanCopulas 175
10.3.2Gumbel–HougaardCopula 176
x Contents
10.3.3FrankCopula 177
10.4EstimationofDependence 180
10.5SimulationStudies 181
10.6CaseStudywithRCodes 184
11ConclusionsandFutureDirections 187
11.1BriefOverview 187
11.2ConcludingRemarks 187
11.2.1LargeSampleSizesforFlexibleModels 187
11.2.2AccurateEstimation 188
11.2.3GoodDesignsBeforeDataAnalysis 188
11.3FutureDirections 189
11.3.1WeibullLifetimeDistributionwithThresholdParameter 189
11.3.2FrailtyModels 189
11.3.3OptimalDesignofSSALTswithMultipleStressLevels 189
11.3.4ComparisonofCSALTsandSSALTs 190
AppendixADerivationof Hi (a, b) 191
AppendixBObservedInformationMatrix 193
AppendixCNon-IdentifiableParametersforSSALTsUnderWeibull Distribution 197
AppendixDOptimalDesignUnderWeibullDistributionswithFixed ��1 199
AppendixEConditionalExpectationsforCompetingRisksModelUnder ExponentialDistribution 201
AppendixFKendall’sTauforFrankCopula 205
Bibliography 207
AuthorIndex 217
SubjectIndex 221
Preface Lifetimeinformationobtainedfromone-shotdevicesisverylimitedastheentire dataareeitherleft-orright-censored.Forthisreason,theanalysisofone-shot devicetestingdataposesaspecialchallenge.Thisbookprovidesseveralstatisticalinferentialmethodsforanalyzingone-shotdevicelifetimedataobtainedfrom acceleratedlife-testsandalsodevelopsoptimaldesignsfortwomainstreamacceleratedlife-tests–constant-stressandstep-stressacceleratedlife-tests–thatare commonlyusedinreliabilitypractice.Thediscussionsprovidedinthebookwould enablereliabilitypractitionerstobetterdesigntheirexperimentsfordatacollectionfromefficientacceleratedlife-testswhentherearebudgetconstraintsinplace. Thisisimportantfromestimationandpredictionpointofviewassuchoptimal designswouldresultinasaccurateaninferenceaspossibleundertheconstraints imposedonthereliabilityexperiment.Moreover,Rcodesarepresentedwithin eachchaptersothatuserscantryoutperformingtheirownanalysisonone-shot devicetestingdata.
Inaddition,theinferentialmethodsandtheproceduresforplanningacceleratedlife-testsdiscussedinthisbookarenotonlylimitedtoone-shotdevices alonebutalsocanbeextendednaturallytoacceleratedlife-testswithperiodic inspections(interval-censoring)andthosewithcontinuousmonitoringand censoring(right-censoring).Thebookfinallyconcludesbyhighlightingsome importantissuesandproblemsthatareworthconsideringforfurtherresearch. Thismaybeespeciallyusefulforresearchscholarsandnewresearchersinterested intakingonthisinterestingandchallengingareaofresearchinreliabilitytheory andpractice.
Itispossiblethatsomepertinentresultsorreferencesgotomittedinthisbook, andweassureyouthatitisduetoinadvertencyonourpartandnotdueto scientificantipathy.Wewillappreciategreatlyifthereadersinformusofany corrections/omissions,oranycommentspertinenttoanyofthediscussionsin thebook!
OursincerethanksgototheentireWileyteam,Ms.MindyOkura-Marszycki, Ms.KathleenSantoloci,andMr.BrettKurzman,fortakinggreatinterestinthis projectfromdayone,foralltheirhelpandencouragementduringthewhole course,andfortheirfineassistanceduringthefinalproductionstageofthebook. Ourthanksalsogotoourresearchcollaboratorsandgraduatestudentsfortheir incisivecommentsandqueries,whichalwaysbenefitedusgreatlyandhelped clarifysomeofourownideas!WeexpressoursincereappreciationtoMs.Elena MariaCastillaGonzalez,adoctoralstudentofProfessorLeandroPardointhe DepartmentofStatisticsandOperationsResearchatComplutenseUniversity ofMadrid,Spain,forhercarefulreadingofChapter5andalsoforsharingwith ussomeRcodesthatshehaddevelopedconcerningrobustinferentialmethods forone-shotdevicetestanalyses.Lastbutnotleast,ourspecialthanksgotoour familiesfortheirpatienceandunderstanding,andforprovidingconstantsupport andencouragementduringourworkonthisbook!
Finally,thefirstauthor(NB)wishestostatetohisolderdaughter,Ms.Sarah Balakrishnan,thatthoughshelostoutongettinghisVolvocarduetoamajor caraccident,sheshouldbeheartenedbythefactthattheaccidentresultedinthe germinationofhisinterestandideasonone-shotdevices(airbags),andultimately thisbooksolelydedicatedtothetopic!
July,2020
NarayanaswamyBalakrishnan ManHoLing HonYiuSo
Thisbookisaccompaniedbyacompanionwebsite:
www.wiley.com/go/Balakrishnan/Accelerated_Life_Testing
TheStudentcompanionsitewillcontainthecodesandcasestudies.
One-ShotDeviceTestingData 1.1BriefOverview One-shotdevicetestingdataanalyseshaverecentlyreceivedgreatattentionin reliabilitystudies.Theaimofthischapteristoprovideanoverviewonone-shot devicetestingdatacollectedfromacceleratedlife-tests(ALTs).Section1.2surveys typicalexamplesofone-shotdevicesandassociatedtestsinpracticalsituations. Section1.3describesseveralpopularALTs,whileSection1.4providessome examplesofone-shotdevicetestingdatathataretypicallyencounteredinreliabilityandsurvivalstudies.Finally,Section1.5detailssomerecentdevelopments onone-shotdevicetestingdataanalysesandassociatedissuesofinterest.
1.2One-ShotDevices Valisetal.(2008)definedone-shotdevicesasunitsthatareaccompaniedby anirreversiblechemicalreactionorphysicaldestructionandcouldnolonger functionproperlyafteritsuse.Manymilitaryweaponsareexamplesofone-shot devices.Forinstance,themissionofanautomaticweapongetscompleted successfullyonlyifitcouldfirealltheroundsplacedinamagazineorinammunitionfeedbeltwithoutanyexternalintervention.Suchdeviceswillusuallyget destroyedduringusualoperatingconditionsandcanthereforeperformtheir intendedfunctiononlyonce.
ShakedandSingpurwalla(1990)discussedthesubmarinepressurehulldamage problemfromaBayesianperspectiveandassessedtheeffectofvariousstrengths ofunderwatershockwavescausedbyeitheranucleardeviceorachemicaldevice ontheprobabilityofdamagetoasubmarinepressurehull.Arecordismadeof whetheracopyofadiminutivemodelofasubmarinepressurehullisdamaged ornot,andaspecificstrengthoftheshockwaveonthemodel.Fanetal.(2009)
AcceleratedLifeTestingofOne-shotDevices:DataCollectionandAnalysis, FirstEdition. NarayanaswamyBalakrishnan,ManHoLing,andHonYiuSo. ©2021JohnWiley&Sons,Inc.Published2021byJohnWiley&Sons,Inc. CompanionWebsite:www.wiley.com/go/Balakrishnan/Accelerated_Life_Testing
consideredelectro-explosivedevicesinmilitaryapplications,whichinducta currenttoexciteinnerpowderandmakethemexplode.Naturally,wecannot adjudgethefunctioningconditionoftheelectro-explosivedevicefromitsexterior, butcanonlyobserveitbydetonatingitdirectly.Afterasuccessfuldetonation, thedevicecannotbeusedanymore;ifthedetonationbecomesafailure,wewill alsonotknowwhenexactlyitfailed.Nelson(2003)describedastudyofcrack initiationforturbinewheels.Eachofthe432wheelswasinspectedonceto determinewhetherithadstartedtocrackornot.Newby(2008)providedsome otherexamplesofone-shotdevices,suchasfireextinguishersormunitions.A fulltestwouldrequiretheuseoftheconsidereddevicesand,therefore,their subsequentdestruction.Thetestcarriedoutwouldshowwhetheradeviceisstill inasatisfactorystate,orhasfailedbythatinspectiontime.
One-shotdevicetestingdataalsoariseindestructiveinspectionprocedures, whereineachdeviceisallowedforonlyasingleinspectionbecausethetestitself resultsinitsdestruction.Morris(1987)presentedastudyof52Li/SO2 storage batteriesunderdestructivedischarge.Eachbatterywastestedatoneofthree inspectiontimesandthenclassifiedasacceptableorunacceptableaccordingtoa criticalcapacityvalue.
Ideally,reliabilitydatawouldcontainactualfailuretimesofalldevicesplaced ontest(assuming,ofcourse,theexperimentercouldwaituntilalldevicesfail), sothattheobservedfailuretimescanrevealthefailurepatternovertime,andwe couldthenestimatethereliabilityofthedevicereasonably.But,inpractice,many life-testswouldgetterminatedbeforealltheunitsfail.Suchanearlystoppageof thelife-testbytheexperimentermaybeduetocostortimeconstraintsorboth. Thiswouldresultinwhatiscalledas“right-censoreddata”becausetheexactfailuretimesoftheunfaileddevicesareunknown,butallweknowisthatthefailure timesofthosedevicesarelargerthantheterminationtime.Considerableliteratureexistsonstatisticalinferenceforreliabilitydataunderright-censoring;for exampleonemayrefertothebooksbyCohen(1991),BalakrishnanandCohen (1991),andNelson(2003).
Moreover,whennondestructiveandperiodicinspectionsarecarriedondevices, theirexactfailuretimeswillnotbeobserved,buttheintervalswhereinthefailures occurredwillonlybeavailable.Ifafailureisobservedbythefirstinspection, thenitisknownthatthefailuretimeofthedeviceislessthanthefirstinspection time,resultingin“left-censoring.”Similarly,ifafailureisobservedbetweentwo consecutiveinspectiontimes,thenitisknownthatthefailuretimeisbetween thesetwocorrespondinginspectiontimes,resultingin“interval-censoring.” Finally,thefailuretimesofallsurvivingunitsatthefinalinspectiontimeare right-censoredastheirexactfailuretimeswillnotbeobserved.Exactfailure timescanonlybeobservedfromalife-testingexperimentwithcontinuous
1.3AcceleratedLife-Tests
surveillance.Theperiodicinspectionprocesswithnondestructiveevaluation wouldactuallyprovideareasonableapproximationtofailuretimesofdevices undertest,especiallywhentheinspectiontimeintervalsareshort,eventhough theprecisionofinferencewillbelessinthiscase.
Itisusefultonotethatinalltheprecedingexamplesofone-shotdevices,we willnotobservetheactuallifetimesofthedevices.Instead,wewouldonlyobserve eitherasuccessorafailureattheinspectiontimes,andsoonlythecorresponding binarydatawouldbeobserved,consequentlyresultinginlesspreciseinference. Inthismanner,one-shotdevicetestingdatadifferfromtypicaldataobtainedby measuringlifetimesinstandardlife-testsand,therefore,posesauniquechallenge inthedevelopmentofreliabilityanalysis,duetothelackoflifetimeinformation beingcollectedfromreliabilityexperimentsonsuchone-shotdevices.Ifsuccessful testsoccur,itimpliesthatthelifetimesarebeyondtheinspectiontimes,leadingto right-censoring.Ontheotherhand,thelifetimesarebeforetheinspectiontimes, leadingtoleft-censoring,iftestsresultinfailures.Consequently,alllifetimesare eitherleft-orright-censored.Insuchasettingofthelifetimedata,Hwangand Ke(1993)developedaniterativeproceduretoimprovetheprecisionofthemaximumlikelihoodestimatesforthethree-parameterWeibulldistributionandto evaluatethestoragelifeandreliabilityofone-shotdevices.Somemoreexamples ofone-shotdevicesintheliteratureincludemissiles,rockets,andvehicleairbags; see,forexample,BainandEngelhardt(1991),Guoetal.(2010),andYunetal. (2014).
1.3AcceleratedLife-Tests Asone-shotdevices(suchasammunitionorautomobileairbags)areusuallykept foralongtimeinstorageandrequiredtoperformitsfunctiononlyonce,the reliabilityrequiredfromsuchdevicesduringtheirnormaloperatingconditions wouldnaturallybehigh.So,itwouldbehighlyunlikelytoobservemanyfailuresontestsundernormaloperatingconditionswithinashortperiodoftime. Thisrenderstheestimationofreliabilityofdevicestobeachallengingproblem fromastatisticalpointofview.Inthisregard,ALTscouldbeutilizedtomitigate thisproblem.InALTs,devicesaresubjecttohigher-than-normalstresslevelsto induceearlyfailures.Inthisprocess,morefailurescouldlikelybeobtainedwithin alimitedtesttime.Astheprimarygoaloftheanalysisistoestimatethereliability ofdevicesundernormaloperatingconditions,ALTmodelswouldthentypically extrapolate(fromthedataobtainedatelevatedstresslevels)toestimatethereliabilityundernormaloperatingconditions.ALTsareknowntobeefficientincapturingvaluablelifetimeinformation,especiallywhenthereisaneedtoshorten
thelife-testingexperiment.Forthisreason,ALTshavebecomepopularandare commonlyadoptedinmanyreliabilityexperimentsinpractice.Onemayreferto thedetailedreviewspresentedbyNelson(1980),CramerandKamps(2001),Pham (2006),andMeekerandEscobar(2014),andtheexcellentbooklengthaccountprovidedbyNelson(2009).
Constant-stressacceleratedlife-tests(CSALTs)andstep-stressaccelerated life-tests(SSALTs)aretwopopularALTplansthathavereceivedgreatattention intheliterature.UnderaCSALT,eachdevicegetstestedatonlyoneprespecified stresslevel.Tomentionafewrecentworks,forexample,Wangetal.(2014) consideredCSALTswithprogressivelyType-IIrightcensoredsamplesunder Weibulllifetimedistribution;forpertinentdetailsonprogressivecensoring, seeBalakrishnan(2007)andBalakrishnanandCramer(2014).Wang(2017) discussedCSALTswithprogressiveType-IIcensoringunderalowertruncated distribution.Linetal.(2019)studiedCSALTsterminatedbyahybridType-I censoringschemeundergenerallog-location-scalelifetimedistributions.SSALTs areanalternativetoapplystresstodevicesinawaythatstresslevelswillincrease atprespecifiedtimesstep-by-step.ForSSALTs,therearethreefundamental modelsfortheeffectofincreasedstresslevelsonthelifetimedistributionofa device:ThetamperedrandomvariablemodelproposedbyDeGrootandGoel (1979),thecumulativeexposuremodelofSedyakin(1966)andNelson(1980); seealso(NikulinandTahir,2013),andthetamperedfailureratemodelproposed byBhattacharyyaandSoejoeti(1989).AllthesemodelsofSSALTshavebeen discussedextensivelybymanyauthors.Gouno(2001)analyzeddatacollected fromSSALTsandpresentedanoptimaldesignforSSALTs;seealsoGouno (2007).ZhaoandElsayed(2005)analyzeddataonthelightintensityoflight emittingdiodescollectedfromSSALTswithfourstresslevelsunderWeibulland log-normaldistributions.Forthecaseofexponentiallifetimedistribution,by consideringasimpleSSALTunderType-IIcensoring,Balakrishnanetal.(2007) developedexactlikelihoodinferentialmethodsforthemodelparameters;see alsoBalakrishnan(2008)fordetails,whileXiongetal.(2006)consideredthe situationwhenthestresschangesfromalow-levelstresstoahigh-levelstressat arandomtime.
1.4ExamplesinReliabilityandSurvivalStudies
1.4.1Electro-ExplosiveDevicesData
Fanetal.(2009)considereddata,presentedinTable1.1,on90electro-explosive devicesundervariouslevelsoftemperatureatdifferentinspectiontimes.Ten devicesundertestateachconditionwereinspectedtoseewhethertherewereany
Table1.1 Failurerecordsonelectro-explosivedevicesunderCSALTs withtemperature(K).
Source:Fanetal.(2009).
failuresornotateachinspectiontimeforeachtemperaturesetting.Thesedata werethenusedtoestimatethereliabilityofelectro-explosivedevicesatdifferent missiontimesunderthenormaloperatingtemperature.
1.4.2GlassCapacitorsData Zelen(1959)presenteddatafromalife-testofglasscapacitorsatfour higher-than-usuallevelsoftemperatureandtwolevelsofvoltage.Ateach oftheeightcombinationsoftemperatureandvoltage,eightitemsweretested. Weadoptthesedatatoformone-shotdevicetestingdatabytakingtheinspection times(hours)as �� ∈{300,350, 400,450}, whicharesummarizedinTable1.2. Thesedatawerethenusedtoestimatethemeanlifetimeofglasscapacitorsfor 250Vand443Ktemperature.
1.4.3SolderJointsData Lauetal.(1988)considereddataon90solderjointsunderthreetypesofprinted circuitboards(PCBs)atdifferenttemperatures.Thelifetimewasmeasuredasthe numberofcyclesuntilthesolderjointfailed,whilethefailureofasolderjointis definedasa10%increaseinmeasuredresistance.Asimplifieddatasetisderived fromtheoriginaloneandpresentedinTable1.3,wheretwostressfactorsconsideredaretemperatureandadichotomousvariableindicatingifthePCBtypeis “copper-nickel-tin”ornot.
Table1.2 FailurerecordsonglasscapacitorsunderCSALTswithtwostress factors:temperature(K)andvoltage(V).
Source:Zelen(1959).
Table1.3 FailurerecordsonsolderjointsunderCSALTswithtemperature(K)anda dichotomousvariableindicatingifthePCBtypeis“copper-nickel-tin(CNT)”ornot.
Source:Lauetal.(1988).
1.4.4Grease-BasedMagnetorheologicalFluidsData
Zhengetal.(2018)studiedgrease-basedmagnetorheologicalfluidsunderSSALTs withfourlevelsoftemperatureandobservedwhethertheirviscositiesorshear stressesdecreasedbymorethan10%aftertests.Twentysamplesofgrease-based magnetorheologicalfluidsweresubjecttohigher-than-normaloperatingtemperature.Then,eachsamplewasinspectedonlyonceandonlywhetherithadfailedor notattheinspectiontimewasobserved,andnottheactualfailuretime.Thedata collectedinthismanner,presentedinTable1.4,werethenusedtoestimatethe meanlifetimeofgrease-basedmagnetorheologicalfluidsunderthenormaloperatingtemperature.
1.4.5MiceTumorToxicologicalData Itisimportanttopointoutthatone-shotdevicetestingdataarisefromdiverse fieldsbeyondreliabilityengineering,suchasinmicetumorstudiesfrom tumorigenicityexperiments;seeKodellandNelson(1980).Insuchastudy,each mousereceivedaparticulardosageofbenzidinedihydrochlorideinitsdrinking waterandwaslatersacrificedtodetectwhethersometumorshaddevelopedby thenornot.Tumorpresencecanbedetectedonlyatthetimeofmouse’ssacrifice ornaturaldeath.ThesedataaresummarizedinTable1.5.Thedatacollectedin thisformwerethenusedtomeasuretheimpactofthechemicaldosageonthe riskoftumordevelopment.
1.4.6ED01ExperimentData LindseyandRyan(1993)describedexperimentalresultsconductedbyNational CenterforToxicologicalResearchin1974.3355outof24000femalemicewererandomizedtoacontrolgrouporgroupsthatwereinjectedwithahighdose(150ppm) ofaknowncarcinogen,called2-AAF,todifferentpartsofthebodies.Theinspectiontimesonthemicewere12,18,and33monthsandtheoutcomesofmicewere deathwithouttumor(DNT)anddeathwithtumor(DWT),andsacrificedwithouttumor(SNT)andsacrificedwithtumor(SWT).Balakrishnanetal.(2016a), intheiranalysis,ignoredtheinformationaboutpartsofmousebodieswherethe drugswereinjectedandcombinedSNTandSWTintoonesacrificedgroup,and denotedthecauseofDNTasnaturaldeathandthecauseofDWTasdeathdueto cancer.ThesedataaresummarizedinTable1.6.Theythenestimatedthechance ofdeathwithouttumor.
1.4.7SerialSacrificeData Lingetal.(2020)wereprimarilyconcernedwiththedata(Berlinetal.,1979),presentedinTable1.7,onthepresenceorabsenceoftwodiseasecategories–(a)
Table1.4 Failurerecordsongrease-basedmagnetorheologicalfluidsunderSSALTswith temperature(K).
Source:Zhengetal.(2018).
Table1.5 Thenumberofmicesacrificed,withtumorfromtumorigenecityexperiments data.
Source:KodellandNelson(1980).
thymiclymphomaand/orglomerulosclerosisand(b)allotherdiseases–foran irradiatedgroupof343femalemicegiven �� -radiationandacontrolgroupof361 radiation-freefemalemicetostudytheonsettimeandtherateofdevelopmentof radiation-induceddisease.Allofthemiceinbothgroupsweresacrificedatvarious times,withthepresenceofadiseaseindicatingthatthediseaseonsetoccurred beforesacrifice,whiletheabsenceofadiseaseindicatingthatthediseaseonset wouldoccuraftersacrifice.
Table1.6 Thenumberofmicesacrificed,diedwithouttumor,anddiedwithtumorfrom theED01experimentdata.
Numberofmice
Test
Source:LindseyandRyan(1993).
Table1.7 Serialsacrificedataonthepresenceorabsenceoftwodiseasecategories:(a) thymiclymphomaand/orglomerulosclerosisand(b)allotherdiseases.
Numberofmice
Test
Source:Berlinetal.(1979).
1.5RecentDevelopmentsinOne-ShotDeviceTesting Analysis Wenowprovideabriefreviewofsomerecentdevelopmentsonone-shotdevice testingdataanalysesunderALTs.ForCSALTs,Fanetal.(2009)comparedthree differentpriordistributionsintheBayesianapproachformakingpredictionson thereliabilityatamissiontimeandthemeanlifetimeofelectro-explosivedevices undernormaloperatingconditions.Inaseriesofpapers,BalakrishnanandLing (2012a,b,2013,2014a)developedexpectation-maximization(EM)algorithms forthemaximumlikelihoodestimationofmodelparametersbasedonone-shot devicetestingdataunderexponential,Weibullandgammalifetimedistributions. Inadditiontoparameterestimation,differentmethodsofconfidenceintervalsfor themeanlifetimeandthereliabilityatamissiontimeundernormaloperating conditionshavealsobeendiscussedbytheseauthors.Themaximumlikelihood estimationaswellasassociatedtestsofhypotheses,thougharemostefficient whentheassumedmodelisindeedthetruemodel,areknowntobenon-robust whentheassumedmodelisviolated,forexample,bythepresenceofsome outlyingvaluesinthedata.Withthisinmind,weightedminimumdensitypower divergenceestimatorshaverecentlybeendevelopedforone-shotdevicetesting dataunderexponential,Weibull,andgammadistributions(Balakrishnanetal., 2019a,b,2020a,b).
Someotherimportantinferentialaspects,apartfromtheworksontheestimationofmodelparametersandhypothesistestsdescribedabove,havealsobeen addressedbyanumberofauthors.BalakrishnanandLing(2014b),forinstance, havedevelopedaproceduretoobtainCSALTplanswhentherearebudgetconstraintsfortestingone-shotdevices.LingandBalakrishnan(2017)alsostudied modelmis-specificationeffectsonone-shotdevicetestingdataanalysesbetween Weibullandgammadistributions,whileBalakrishnanandChimitova(2017)conductedcomprehensivesimulationstudiestocomparetheperformanceofseveral goodness-of-fittestsforone-shotdevicetestingdata.
Intheframeworkofcompetingrisksanalysis,Balakrishnanetal.(2015,2016a,b) discussedtheanalysisofone-shotdevicetestingdatawhenthedevicescontain multiplecomponentsandhencehavingmultiplefailuremodes.Anotherextensionthathasbeenprovidedforone-shotdevicetestingisbyLingetal.(2016)who havedevelopedproportionalhazardsmodelsforanalyzingsuchdata.Optimal
SSALTplansforone-shotdevicetestingexperimentwithlifetimesfollowingexponentialandWeibulldistributionshavebeendiscussedbyLing(2019)andLingand Hu(2020).
PanandChu(2010)haveinvestigatedtwo-andthree-stageinspectionschemes forassessingone-shotdevicesinseriessystemsofcomponentshavingWeibulllifetimedistributions.Finally,ChengandElsayed(2016–2018)haveexaminedseveral approachestomeasurethereliabilityofone-shotdeviceswithmixtureofunits undervariousscenariosandhavepresentedreliabilitymetricsofsystemswith mixturesofnonhomogeneousone-shotunitssubjecttothermalcyclicstressesand furtheroptimaloperationaluseofsuchsystems.
Inthechaptersthatfollow,weshallelaborateonallthesedevelopmentsand alsohighlighttheirapplications.
LikelihoodInference 2.1BriefOverview Likelihoodinferenceisoneoftheclassicalmethodsfortheestimationofmodel parameters,anditreliesonthemaximizationoflikelihoodfunctionofanassumed modelbasedontheobserveddata.Inthischapter,wedetailtheprocedureof findingthemaximumlikelihoodestimates(MLEs)ofmodelparameters,mean lifetime,andreliabilityundernormaloperatingconditionsonthebasisofone-shot devicetestingdataunderconstant-stressacceleratedlife-tests(CSALTs)fordifferentlifetimedistributions.Severalassociatedintervalestimationmethodsarealso discussedforalllifetimeparametersofinterest.Thediscussionsprovidedhereare primarilyfromtheworkscarriedoutbyBalakrishnanandLing(2012a,b,2013, 2014a).Then,theresultsofasimulationstudyevaluatingtheperformanceofthe developedpointandintervalestimationmethodsarepresented.Finally,twodata setspresentedinthelastchapterareusedtoillustratealltheinferentialresults discussedhere.
2.2UnderCSALTsandDifferentLifetimeDistributions SupposeCSALTsareperformedon I groupsofone-shotdevices.Supposefor i = 1, 2, , I , inthe ithtestgroup, Ki one-shotdevicesaresubjectto J typesofacceleratingfactors(suchastemperature,humidity,voltage,andpressure)atstresslevels xi =(xi1 , xi2 , , xiJ ) andgetinspectedattime ��i fortheircondition.Thecorrespondingnumbersoffailures, ni , arethenrecorded.Thedatathusobservedcan besummarizedasinTable2.1.
Fornotationalconvenience,weuse z ={��i , Ki , ni , xi , i = 1, 2, , I } todenotethe observeddata,and �� forthemodelparameter.Supposethelifetimeinthe ithtest grouphasadistributionwithcumulativedistributionfunction(cdf) F (t; ��i ), with
AcceleratedLifeTestingofOne-shotDevices:DataCollectionandAnalysis, FirstEdition. NarayanaswamyBalakrishnan,ManHoLing,andHonYiuSo. ©2021JohnWiley&Sons,Inc.Published2021byJohnWiley&Sons,Inc. CompanionWebsite:www.wiley.com/go/Balakrishnan/Accelerated_Life_Testing
Table2.1 One-shotdevicetestingdataunderCSALTswithmultipleacceleratingfactors, variousstresslevels,anddifferentinspectiontimes.
TestInspection groupStresslevelstimeNumberoftesteddevicesNumberoffailures
Theobservedlikelihoodfunctionisthengivenby
) isthereliabilityatinspectiontime ��i and C isthe normalizingconstant.Wethenmakeuseofthedatatodeterminesomelifetime characteristicsofdevices,fromthelikelihoodfunctionin(2.1),undernormal operatingconditions x0 =(x01 , x02 , … , x0J ), suchasthemodelparameter, ��, mean lifetime, ��, andreliabilityatmissiontime t, R(t)
2.3EM-Algorithm Likelihoodinferenceonone-shotdevicetestingdatahasbeendiscussedextensivelyformanyprominentlifetimedistributions;seeBalakrishnanandLing (2012a,b,2013,2014a).Inthissection,wedescribetheexpectation-maximization (EM)frameworkfordeterminingtheMLEsofallparametersofinterest.
Inone-shotdevicetesting,asmentionedearlier,noactuallifetimesare observed,and,assuch,allobserveddataarecensored.EMalgorithmisknown tobeaconvenientandefficientmethodforestimatingmodelparametersinthe presenceofcensoring;see,forexample,McLachlanandKrishnan(2008)forall pertinentdetailsconcerningthismethod,itsvariations,andapplications.This methodhasnowbecomeastandardmethodofmodel-fittinginthepresenceof missingdata.TheEMalgorithminvolvestwostepsineachiterationofthenumericalmethodofmaximizingthelikelihoodfunctionsuchastheonepresentedin (2.1):expectation-step(E-step)inwhichthemissingdataareapproximatedby theirexpectedvalues,andmaximization-step(M-step)inwhichthelikelihood function,withimputedvaluesreplacingthemissingdata,getsmaximized.
Supposethelifetimesofdevicesfollowadistributionwithprobabilitydensity function(pdf) f (t; ��). Wefirstnotethatthelog-likelihoodfunction,basedonthe completedata,is
Inthe mthstepoftheiterativeprocess,theobjectivethenistoupdatetheestimateofparameter �� bythevaluethatmaximizesthefunction
basedonthecurrentestimate ��(m) Thecurrentestimate ��(m) isusedinobtaining theupdatedestimate ��(m+1) intheM-step,andthentheconditionalexpectation Q(��, ��(m+1) ) in(2.3)isobtainedbasedon ��(m+1) intheE-step.Thesetwostepsare thenrepeateduntilconvergenceisachievedtoadesiredlevelofaccuracy.Itis evidentthatweactuallysolvetheincompletedataproblemofmaximizingthelikelihoodfunctionin(2.1)byfirstapproximatingthemissingdataandthenusingthe approximatedvaluestofindtheestimateoftheparametervectorasthesolution forthecompletedataproblem.
However,inthemaximizationstep,aclosed-formsolutionmaynotalways befound,andinsuchacase,aone-stepNewton–Raphsonmethodcanbe employedforthispurpose.Thiswouldrequirethesecond-orderderivativesofthe log-likelihoodfunctionwithrespecttothemodelparameters.Inthepresentcase, letusintroduce
and
Then,theupdatedestimateobtainedthroughtheone-stepNewton–Raphson methodisgivenby
Inadditiontoestimatesofmodelparameters,ourinterestmayalsobeon somelifetimecharacteristics,suchasmeanlifetimeandreliabilityundernormal operatingconditions.Asthesequantitiesarefunctionsofmodelparameters,their estimatescanbereadilyfoundbypluggingintheMLEsofmodelparameters intotherespectivefunctions,andthenthedeltamethodcanbeemployedto determinethecorrespondingstandarderrorsaswell;see,forexample,Casella andBerger(2002).
2.3.1ExponentialDistribution Letusfirststartwiththeexponentialdistribution,whichisoneofthemostpopular lifetimedistributions;see,forexample,Johnsonetal.(1994)andBalakrishnan andBasu(1995)fordevelopmentsonvariousmethodsandapplicationsofthe exponentialdistribution.Inthiscase,Fanetal.(2009)andBalakrishnanandLing (2012a,b)studiedinferentialmethodsforone-shotdevicetestingdataunderexponentialdistributionwithsingleandmultipleacceleratingfactors.Letususe E to denotetheexponentialdistributioninwhatfollows.Inthiscase,inthe ithtest group,thecdfandpdfoftheexponentialdistribution,withrateparameter ��i > 0, aregivenby
respectively.Itsmeanlifetimeandreliabilityatmissiontime t undernormaloperatingconditions x0 =(x01 , x02 , … , x0J ) arethengivenby
and
(t)= 1 FE (t; ��0 )=
respectively.Asitisnecessarytoextrapolatethedataobservedfromtheelevated stresslevels xi =(xi1 , xi2 , … , xiJ ), tolifetimecharacteristicsofdevicesundernormaloperatingconditions x0 =(x01 , x02 , … , x0J ), therateparameter ��i in(2.7)gets relatedtothestresslevelsinalog-linearlinkfunctionoftheform,with xi0 = 1, ln(��
where ��E =(e0 , e1 , , eJ ) nowbecomesthenewmodelparameter. Itisworthmentioningthatmanywell-knownstress-ratemodels,suchas Arrhenius,powerlaw,inversepowerlaw,andEyringmodels,areallspecialcases ofthelog-linearlinkfunctionin(2.9),withsuitabletransformationsonthestress variables;see,forexample,WangandKececioglu(2000).
UndertheEMframework,thelog-likelihoodfunctionbasedonthecomplete data,obtainedreadilyfrom(2.8),isgivenby
andtheconditionalexpectationin(2.3)becomes
where A(m) i = E[Ti |z,�� (m) E ] canbecomputedintheE-step,asdescribedbelow. Thefirst-orderderivativeswithrespecttothemodelparameters,requiredfor themaximizationofthequantity Q(��E , ��(m) E ) in(2.10),arethengivenby
Aclosed-formsolutionisnotavailablefor(2.11),however.So,intheM-stepof theEMalgorithm,weemploytheone-stepNewton–Raphsonmethod,asmentionedearlier.Thisrequiresthesecond-orderderivativesof(2.10)withrespectto themodelparameters,whicharegivenby
Next,theconditionalexpectationrequiredintheE-stepiseasytoderiveasit onlyinvolvesthefailedandunfaileddevices.Forthefaileddevices,theproportion is ni ∕Ki ,andthelifetimesinthiscaseareright-truncatedattheinspectiontime ��i (astheycannotbemorethan ��i ).Similarly,fortheunfaileddevices,theproportion is1 ni ∕Ki ,andthelifetimesareleft-truncatedattheinspectiontime ��i (asthey cannotbelessthan ��i ).Consequently,weobtaintheconditionalexpectationof Ti , giventheobserveddataandthecurrentestimate ��(m) E of ��E , asfollows: A(m) i = E[Ti |z, ��(m) E ] = ( ni Ki )
i 0 tfE (t; ��(m) E )dt FE (��i ; ��(m) E ) + (1 ni Ki
where ��(m) i = exp(∑J j=0 e(m) j xij ) and ̂ F (��i )= ni ∕Ki istheempiricalprobabilityoffailurebyinspectiontime ��i
WhentheMLEsofthemodelparameters ̂ ��E =( ̂ e0 , ̂ e1 , … , ̂ eJ ) aredetermined,we canreadilyhavetheMLEsofmeanlifetimeandreliabilityatmissiontime t under
normaloperatingconditions x0 =(x01 , x02 , … , x0J ), forexample,tobe
and
respectively.
2.3.2GammaDistribution Astheexponentialdistributionwouldonlybesuitableformodelingconstant hazardrate,itwouldbenaturaltoconsiderthelifedistributiontobegammaas itpossessesincreasing,decreasing,andconstanthazardrateproperties;see,for example,Johnsonetal.(1994).Moreover,itincludesexponentialdistributionas aparticularcase.Forthisreason,BalakrishnanandLing(2014a)extendedthe worktogammalifetimedistributionwithmultipleacceleratingfactorswithboth scaleandshapeparametersvaryingoverstressfactors.
Letususe G todenotegammadistributioninwhatfollows.Specifically,inthe ithtestgroup,weassumethelifetimestofollowagammadistributionwithshape parameter ��i > 0andscaleparameter ��i > 0, withcdfandpdfgivenby
and
respectively,where
∕Γ(z) isthelowerincomplete gammaratioand Γ(z)= ∫ ∞ 0 x z 1 exp(−x )dx isthecompletegammafunction.The meanlifetimeandreliabilityatmissiontime t undernormaloperatingconditions x0 =(x01 , x02 , … , x0J ) are,respectively,givenby
where Γ(z, s)= 1 �� (z, s) istheupperincompletegammaratio.Forextrapolatingthedataobservedfromtheelevatedstresslevels xi =(xi1 , xi2 , … , xiJ )
tolifetimecharacteristicsofdevicesundernormaloperatingconditions x0 = (x01 , x02 , … , x0J ), inthiscase,bothparameters ��i and ��i arerelatedtothestress levelsinlog-linearlinkfunctionsoftheform,with xi0 = 1,
Thus,thenewmodelparameteroftheconsideredmodelis
Here,undertheEMframework,thelog-likelihoodfunction,basedonthecompletedata,isreadilyobtainedfrom(2.14)tobe
Let A(m) i = E[ln(Ti )|z, ��(m) G ] and B(m) i = E[Ti |z, ��(m) G ]. Then,theconditionalexpectationin(2.3)becomes
Thefirst-orderderivativeswithrespecttothemodelparameters,requiredfor themaximizationofthequantity Q(��G , ��(m) G ) in(2.16),aregivenby �� Q(��G , ��(m) G ) �� aj
(
) i
, for j = 0, 1, … , J . Thecorrespondingsecond-orderderivativesare
2 Q(��G , ��(m) G ) �� ap �� aq =− I
=1
2 Q(��G , ��(m) G ) �� bp �� bq =− I
Q(��G , ��(m) G ) �� ap �� bq =− I
xip xiq ��i (��i Ψ′ (��i )+Ψ(��i )+ ln(��i )− A(m) i ),
20 2LikelihoodInference
for p, q = 0, 1, … , J , where Ψ(z)= �� ln Γ(z)∕�� z and Ψ′ (z) arethedigammaand trigammafunctions,respectively;seeAbramowitzandStegun(1972)fora discussiononthecomputationofthesefunctions.
Hereagain,theconditionalexpectationsrequiredintheE-step, A(m) i and B(m) i , onlyinvolvethefailedandunfaileddevices.Forthefaileddevices,theproportion is ni ∕Ki ,andthelifetimesareright-truncatedattheinspectiontime ��i ,whileforthe unfaileddevices,theproportionis1 ni ∕Ki ,andthelifetimesareleft-truncated attheinspectiontime ��i . Consequently,weobtaintheconditionalexpectations, giventheobserveddataandthecurrentestimate ��(m) G of ��G , asfollows:
�� (m) i 0 ln(t)t
(m) i 1 exp(−t)dt Γ(�� (m) i )FG (��i ; ��(m) G ) + (1 ni Ki ) ∫ ∞ ��i ∕�� (m) i ln(t)t�� (m) i 1 exp(−t)dt
Γ(�� (m) i )(1 FG (��i ; ��(m) G )) = ln(�� (m) i )+ H1 (�� (m) i , ��i �� (m) i )(
̂ F (��i ) FG (��i ; ��(m) G ) )
+ {Ψ(�� (m) i )− H1 (�� (m) i , ��i �� (m) i )}( 1
̂ F (��i ) 1 FG (��i ; ��(m) G ) ) and
B(m) i = E[Ti |z, ��(m) G ] = ( ni Ki ) ∫ ��i 0 tfG (t; �� (m) i ,�� (m) i )dt FG (��i ; ��(m) G ) + (1 ni Ki ) ∫ ∞ ��i tfG (t; �� (m) i ,�� (m) i )dt 1 FG (��i ; ��(m) G )
= �� (m) i �� (m) i �� (�� (m) i + 1, ��i �� (m) i )(
̂ F (��i )
FG (��i ; ��(m) G ) )
+ �� (m) i �� (m) i à (�� (m) i + 1, ��i �� (m) i )( 1
̂ F (��i ) 1 FG (��i ; ��(m) G ) ) ,
