用FRCHET分布设计步进应力加速寿命试验最佳时间的图形表示(IJITCS-V6-N12-10)

I.J.InformationTechnologyandComputerScience,2014,12,74-81PublishedOnlineNovember2014inMECS()DOI:10.5815/ijitcs.2014.12.10Copyright©2014MECSI.J.InformationTechnologyandComputerScience,2014,12,74-81GraphicalRepresentationofOptimalTimeforaStep-StressAcceleratedLifeTestDesignUsingFrechetDistributionSanaShahabDepartmentofStatistics&OperationsResearch,AligarhMuslimUniversity,Aligarh,IndiaEmail:[e-mail]-Ul-IslamDepartmentofStatistics&OperationsResearch,AligarhMuslimUniversity,Aligarh,IndiaEmail:[e-mail]—ThearticleprovidesanapproachofgettingoptimaltimethroughgraphforSimplestepstressacceleratedtestofinverseweibulldistribution.Inthisweestimateparametersusingloglinearrelationshipbymaximumlikelihoodmethod.Alongwiththis,asymptoticvarianceandcovariancematrixoftheestimatorsaregiven.ComparisonbetweenexpectedandobservedFisherInformationmatrixisalsoshown.Furthermore,confidenceintervalcoverageoftheestimatorsisalsopresentedforcheckingtheprecessionofestimator.Thisapproachisillustratedwithanexampleusingsoftware.IndexTerms—AcceleratedLifeTesting,Step-Stress,Frechet(InverseWeibull)Distribution,MaximumLikelihood,AsymptoticVariance(AV),OptimalTime,ConfidenceIntervalI.INTRODUCTIONAmoregeneralizedcaseisusedinstepstressforthefulfillmentofalltheapplications.Frechetdistributionisimportantformodelingthestatisticalbehaviorofmaterialsforavarietyofengineeringapplications.Ithandlessensitivecircuitsveryeasilyandisalsousedforoptoelectronicdevicesuchassolarcell,photodiodes,phototransistor,lightemittingdevicesetc.Duetocontinuousimprovementinthetechnology,theproductstodayhavebecomemoreandmorereliablewithmorelife.Itmighttakealongtime,maybeseveralyears,foraproducttofail,whichmakesitdifficultandevenimpossibletoobtainthefailureinformationunderusageconditionforsuchhighlyreliableproducts.Sotogettheinformationaboutthelifetime,asampleoftheseproductsissubjectedtomoresevereoperatingconditionsthannormalonestoobtainitsfailuremode.Thistypeoftestingiscalledtheacceleratedlifetesting(ALT),wheretheproductsareputunderhigherthanusualstressestogetmorefailuredatainshorttime.ThebasicgoalofALTistoproducehighqualityproductatlowcostandlesstime.Thestresscanbeappliedindifferentways.Commonlyusedmethodsareconstantstress,progressivestressandstepstress:Constant-stressALT:Inthistype,stressiskeptataconstantlevelthroughoutthelifeoftestproducts.Someoftheimportantearlyworksinconstant-stresstestcanbefoundinKelpinskiandNelson[1],NelsonandMeeker[2].Progressive-stressALT:Inthistype,stressappliedtoatestproductiscontinuouslyincreasingwithtime.See,BalakrishnanaandHan[3].Step-stressacceleratedlifetesting(SSALT):Inthisstressesareincreasedinstepwisemanneri.e.firstlytheproductissubjectedtoaspecifiedconstantstressS1,foraspecifiedlengthoftime.Ifitdoesnotfail,itissubjectedtoahigherstresslevelS2untilitfails.Sincehigherstressesareusedforbetterresult,soacceleratedtestingmustbeapproachedwithcautiontoavoidintroducingfailuremodesthatwillnotbeencounteredinnormaluse.FormoreInformationaboutALTsonecanconsultNelson[4].Hewasthefirsttoproposethesimplestep-stressscheme,withthecumulativeexposuremodel.ManystudiesregardingSSALTplanninghavebeenperformedbasedontheCEModel,seeXiong[5],Watkins[6],ZhaoandElsayed[7],Balakrishnanelal.[8],YeoandTang[9].MillerandNelson[10]obtainedtheoptimumsimplestep-stressacceleratedlifetestplansforthecasewherethetestunitshaveexponentiallydistributedlifetimes.Baiandothers[11]extendedtheresultsofMillerandNelson[10]tothecaseofcensoring.Khamis-Higgins[12]introducedthestepstressschemeforweibulldistributionusingK-Hmodel.AliandAmmar[13]proposedOptimalDesignofStep-StressLifeTestwithProgressivelytype-IICensoredExponentialData.Tangetal.[14]haveusedalinearcumulativeexposuremodeltoanalyzedatafromaSSALTusing3-parameterWeibulldistribution.BhattacharyyaandSoejoeti[15]developedatamperedfailure-ratemodel.Bhattacharyya[16]alsoderivedanapproachusingaGaussianstochasticprocesswhichwaslatermodifiedandextendedbyDoksumandHoyland[17].ThecumulativeexposuremodeldefinedbyNelson[4]forsimplestep-stresstestingwithlowstressS1andhighstressS2:GraphicalRepresentationofOptimalTimefora75Step-StressAcceleratedLifeTestDesignUsingFrechetDistributionCopyright©2014MECSI.J.InformationTechnologyandComputerScience,2014,12,74-81ttFt0tFtF21where:Fi(t)thecumulativedistributionfunction(c.d.f.)ofthefailuretimeatstressSi,τisthetimetochangestressandτ'isthesolutionofF1(τ)=F2(τ').Onsolvingforτ'weget:12TheReliabilityfunctionofFrechetdistributionis:0,t,texp1)t(RTheremainderofthispaperisorganizedasfollows:InSection2weprovidethesimpleconditionsandassumptionsonwhichwholepaperisbased.NextSection3presentsthemaximumlikelihoodestimators(MLEs)ofmodelaswellasFisherInformationmatrix.Alongwiththisvariance-covariancematrixisalsodiscussed.Section4givestheconfidenceintervaldetailsfollowedbycalculationofoptimaltimewiththehelpofgraphinsection5.Section6explainsthesimulationstudiesforillustratingthetheoreticalresults.Finally,conclusionsar