ACCELERATED LIFE TESTING MODEL BUILDING WITH BOX-C

ACCELERATEDLIFETESTINGMODELBUILDINGWITHBOX-COXTRANSFORMATIONPeihuaQiuChrisP.TsokosSchoolofStatisticsDepartmentofMathematicsUniversityofMinnesotaUniversityofSouthFloridaMinneapolis,MN55455Tampa,FL33620AbstractInacceleratedlifetesting,thenominallifetimeisoftenrelatedtostresslevelsbyanaccel-erationequation.Threeparticularmodelsthathavebeenusedfrequentlyinthepastarethepowerlawmodel,theArrheniusmodelandtheEyringmodel.Inthispaperwesuggestchoosingamodelfromamodelfamilywhichincludesthethreeparticularmodelsasspecialcases.ThisfamilyisdenedbyaBox-Coxtransformationonthestressvariable.Therearetwobenetstousethisproposal:(1)modelttingcouldbetreatedinanuniedway;and(2)thettedmodelismorerobusttomodelassumptions.Wedemonstratethismethodwithsomenumericalexamples.KeyWords:Acceleratedlifetesting,Accelerationequation,Box-Coxtransformation,Residualsumofsquares.1IntroductionInacceleratedlifetesting,productsaretestedunderhigherthanusuallevelsofstressestoshortenthetestingtimeandtogetmorefailures(Nelson1990;MeekerandEscobar1993).Toestimatelifetimesatnormalstresslevelsbasedontheacceleratedlifetestingdataisaprocessofextrapolation.Thisprocessisoftenaccomplishedbyusingapredeterminedaccelerationequationwhichrelatesthelifetimeofproductstothestresslevels.Threeparticularaccelerationequationsthathavebeenusedfrequentlyinthepastarethepowerlawmodel,theArrheniusmodelandtheEyringmodel(Levenbach1957;Thomas1964).1Thepowerlawmodelhasalinearexpression:=a+b[log(S)],where=log(),isthenominallifetime(someparameterofthelifetimedistribution),Sdenotesthestressvariable,aandbarethecoecients.Thismodelisoftenusedtorelateproductlifetopressure-likestresses(e.g.,voltage).Itisgenerallyviewedasbeinganempiricalmodel,butwithlargeamountsofexperimentalverication(seealistofapplicationsofthismodelinSection2.10,Nelson1990).TheArrheniusliferelationship,=a+b=S,iswidelyusedtomodelproductlifeasafunctionoftemperature.Applicationsincludeelectricalinsulationsanddielectrics,batterycells,plastics,etc.Itisarst-orderapproximationtothefollowingEyringmodel:=log(A)log(S)+B=S,whereAandBareconstants.Inmostapplications,A=Sisessentiallyconstantduetothesmallrangeoftemperature,makingtheEyringmodelclosetotheArrheniusrelationship.Theabovethreemodelshavethefollowingcommonstructure:=a+b(S);(1.1)where()issomeprespeciedfunction,aandbarethecoecients.Intheliterature(e.g.,Chapters4and5,Nelson1990),themodelcoecientsareoftenestimatedbytheleastsquares(LS)methodandthemaximumlikelihoodestimation(MLE)method.ByusingtheLSmethod,needstobeestimatedfromtheexperimentaldatarstateachstresslevelandthenmodel(1.1)isttedintheusualway.Model(1.1)isbasedontwoassumptions:(1)thefunction()needstobecompletelyspecied,and(2)therelationshipbetweenand(S)islinear.Ifoneofthesetwoassumptionsisviolatedinaspecicapplication,thenresultsfromtheextrapolationprocedurewillnotbereliable.Thereforeitisemphasizedintheliterature(e.g.,Chapter2,Nelson1990;Chapter18,MeekerandEscobar1998)tofullyunderstandthemechanismoftheapplicationproblemssuchthatappropriatemodelscouldbeidentiedforextrapolation.Itisalsoemphasizedtoverifytheempiricalmodelsovertheentirerangeofthestressvariables.Butitmightnotbeeasytodosoinsomecasesbecausethelifetimeofsomeproductscouldbeextremelylongunderlowstresses.Inthispaper,wemakeanattempttotrytopartiallyovercomethisdicultybyconsideringamoreexiblemodel.Figure1.1demonstratesfourpossiblecases.ThestressSinthesecasesisthetemperatureT.WeconsidersevenTlevels:150oC;200oC;250oC;300oC;350oC;400oCand450oC.Inplot(a),thetruerelationshipbetweenand(T)is=38:3+5(T)+5log((T))where(T)=1=T.Thelinealityassumptionof(1.1)isviolatedinthiscase.(ItisanArrheniusmodeliftheterm25log((T))doesnotexist.)The\+pointsinFigure1.1(a)denotef((Ti);^i)gwhere^iisanestimatorofatstresslevelsTi.Thedottedcurverepresentsthetrueregressionmodel.ThedashedlineisthettedLSlinebyusingtheArrheniusmodel.Thesolidcurvedenotesthettedmodelbyourproposal.Itcanbeseenthatallthreecurves/linesareclosetoeachotherinthedesignrange(Tbetween150oCand450oC).Butwhentheyareusedforextrapolationatnormaltemperaturelevels(say,T100oC),theirdierenceisobvious.ExtrapolationresultsfromourproposalareclosetothetruthwhilethosefromtheArrheniusmodelarefarawayfromthetruth.ThisexampleshowsthattheextrapolationprocedureissensitivetothelinealityassumptioniftheArrheniusmodelisusedinstatisticalanalysisandourproposalpartiallyovercomesthisproblem.Plot(b)demonstratesanothercasethatthetruerelationshipbetweenand(T)islinear.But(T)equalstolog(T)=T0:6insteadof1=T.Thetruemodelis=3:7+18:5(T).Weplotf((Ti);^i)gby\+pointsasbefore.Thedottedstraightlineisthetrueregressionmodel.ThedashedcurverepresentsthettedArrheniusmodel(itdoesnotappeartobestraightinplot(b)becausethescaleusedforthex-axisisbylog(T)=T0:6insteadofbyT).Thesolidcurveisthettedmodelbyourproposal.TheextrapolationresultsbyusingtheArrheniusmodeldonotlookgoodinthiscaseeither.Ontheotherhand,ourmethodstillbehavesreasonablywell.Plots(c)and(d)showanothertwocasesrelatedtothepowerlawmodel.ThetruerelationshipbetweenandTis=20:3+1500