Gupta Amirkabir University of Technology Gonbad Kavous University and Bowling Green State University Abstract The Weibull distribution has received much interest in reliability theory The wellknown maximum likelihood estimators MLE of this fam ily ID: 31950
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404MahdiTeimouriandArjunK.Gupta respectively,forx�, �0,�0.Theparameters, ,andareknownastheshape,scaleandlocationparameters,respectively.Thehazardratefunctioncorrespondingto(1.1)and(1.2)isH(x)= x� �1;(1.3)forx�.So,theWeibulldistributioncanallowfordecreasing,constantandincreasinghazardrates.ThisisoneoftheattractivepropertiesthatmadetheWeibulldistributionsoapplicable.Thenon-centralmomentscorrespondingto(1.1)and(1.2)aregivenbyE(Xr)=rXi=0riir�i�1+r�i ;(1.4)wherer�i� and�()denotesthegammafunction.Manyestimationmethodshavebeenproposedforestimatingtheparame-tersoftheWeibulldistribution.Wemention:maximumlikelihoodestimation(SirvanciandYang,1984),momentsestimation(Cohenetal.,1984;Cran,1988),Bayesianestimation(Tsionas,2000),quantileestimation(WangandKeats,1995),logarithmicmomentestimation(Johnsonetal.,1994)andtheprobabilityweightedmomentestimation(Bartoluccietal.,1999).Themostpopularandthemostef-cientoftheseisthemaximumlikelihoodestimation.2.MainResultsAmongmentionedinferencemethods,asthemostecientonewhichreceivedalotofattentionintheliterature,MLEofWeibullfamilyisderivedbysolvingthenon-linearsetofthreeequationsgivenasfollows:n +nXi=1logxi� �nXi=1xi� logxi� =0;(2.1)�n + nXi=1xi� =0;(2.2)�( �1)nXi=11 xi�+ nXi=1xi� �1=0:(2.3)Theseequationsdonotyieldclosedformsolutionsforparameters.Theorem2.1isusefulforconstructingasimple,consistentandclosedformestimatorfor .Thisestimatorisindependentof. 406MahdiTeimouriandArjunK.Gupta Now,divideandsimultaneouslymultiplyby,respectively,Xandq 12(n�1) n+1thebothsidesof(2.8)toseetheresult.2Corollary2.1.Supposex1;x2;;xnisarandomsamplefrom(1.1)withknownlocationparameter.Letdenotethesamplecorrelationbetweenthexiandtheirranks.LetCVandS,respectively,denotethesamplecoecientofvariationandsamplestandarddeviation.Thenanestimatorfortheshapeparameter, is:^ =�ln2 lnh1� p 3�1 CV� S�1q n+1 n�1i:(2.9)FromJohnsonetal.(1994,p.656),insomecertaincases,itiswellknownthatX(1)=minfX1;X2;;XngisMLEfor.Generally,thisstatisticisaconsistentestimatoroflocationparameter(seeKunduandRaqab,2009).AbetterestimateisX(1)�1=n(seeSirvanciandYang,1984,p.74).Wetakethelatterstatisticasanestimatoroftheunknownparameter,i.e.,welet^=X(1)�1=n.Itcanbeusedforconstructinganewand-independentestimatorofshapeparameter, asfollows.Corollary2.2.Supposex1;x2;;xnisarandomsamplefrom(1.1)withunknownlocationparameter.Letdenotethesamplecorrelationbetweenthexiandtheirranks.LetCV,SandX(1),respectively,denotethesamplecoecientofvariation,samplestandarddeviationandminimumstatistic.Thenanewestimatorfortheshapeparameter, is:^ =�ln2 ln1� p 31 CV�X(1)�1=n S�1q n+1 n�1:(2.10)3.PerformanceAnalysisHereweanalyzetheperformanceofthenewestimatorgivenin(2.10).Forthesakeofsimplicity,letustoconsidertwocases:1�isknownand2�isunknown.Case1:Inthiscase,aftersubtractingfromallthepointsofdataset,theproblemisreducedtoestimatingtheshapeparameteroftwo-parameterWeibulldistribution.Nowthenewestimator(2.9)dependsonlyontheCVstatisticandis:^ =�ln2 lnh1� p 3CVq n+1 n�1i:(3.1) 408MahdiTeimouriandArjunK.Gupta where^ idenotesthevalueofeithernewestimatororMLEinithiteration.First,notethathereafterweconsiderthatisunknownandwecall(2.10)asnewestimator.Toestablishacomprehensivesimulation-basedstudyformeasuringthee-ciencyofnewestimatorcomparisonwiththeMLE,MREiscomputedforsamplesizesof100,500,1000and2000whenisseton5.LargervaluesoftheMREcorrespondtolessecientestimator.Figure1displaystheMREsforasampleofsizen=100,500,1000and2000forsomelevelsof=0.5,5and=0.TheMREsforasampleofsizen=100,500,1000and2000forsomelevelsof=0.5,5and=10areshowninFigure2.Itshouldbenotedthat,here,wesetthenumberofiterationsk,at100.ThefollowingobservationscanbemadefromFigures1and2:1.inallcasesthedi erencebetweenMREoftheNewestimatorandtheMLEisnotsignicant.2.becausetheNewestimatorisscaleinvariant,totally,di erencebetweenMREsoftheNewestimatorandMLEisnotsubjecttothescaleparameter.3.ineachrowofFigures,whenincreasesfrom0.5to5,MREshavenosignicantchanges,totally.4.ineachcolumnofFigures,whennincreasesfrom100to2000,MREsde-creasewithincreasingn,totally.5.whenincreasesfrom0to10,comparingFigures1and2,itturnsoutthatdegreeofdependenceofMREsoftwoestimatorsonisnegligible.FurtherdiscussionsshowthatbothnewestimatorandMLEbehavesthesamewhenbiasisconsideredascriterion.Themodeoftwoestimatorsoccuratorigin.AlthoughtheMLEhashigherpeakthanthenewestimatorinorigin,butsimula-tionsshowthatthesamplerangeofbiasoftheNewestimatorandtheMLEareapproximatelyequal.Also,normalityoftheNewestimatorisveriedevenformsmallsamplesize(heren=100).ThebiasfrequencyhistogramaredepictedinFigure3forsomeselectedlevelsof andwhenlocationparameter,isseton10.Thehistogramisconstructedfrom500points,withaccounttakenofthefactthateachpointisobtainedviatheMLEornewestimatoronthebasisofasampleofsize100generatedfrom(1.1).3.2.ExamplesInthissubsection,weprovidetwodatasetstoshowhowmuchthenewestimatorworkswell.Forthismean,weaddressthereaderstodatasetsisassumedtobedistributedwithWeibulllaw(seeMurthyetal.,2004,pp.83,100).ThedatasetsaregiveninTables1and2asfollows.Theresultsfortting 410MahdiTeimouriandArjunK.Gupta Figure2:MREoftheNewestimatorandMLEforsomelevelsof andwhen=10 OnTheThreeParameterWeibullDistribution411 (MLE)(Newestimator) ( =1,=1)( =1,=1)(MLE)(Newestimator) ( =5,=5)( =5,=5)Figure3:BiasfrequencyoftheNewestimatorandtheMLE forsomelevelsof andthethree-parameterWeibulldistributiontothedatasetsofTables1and2aregiven,respectively,inTables3and4.Itshouldbenotedthat,afterestimating ,^isestimatedthroughaclosedformexpressionderivedby(2.2).AlsoX(1)�1=nisconsideredhereasanestimatorfor.Table1:Dataset1,failuretimesof24mechanicalcomponents 30.9418.5116.6251.5622.8522.3819.0849.5617.1210.6725.4310.2427.4714.7014.1029.9327.9836.0219.4014.9722.5712.2618.1418.84 Table2:Dataset2,lifetimesof20electroniccomponents 0.030.120.220.350.730.791.251.411.521.791.801.942.382.402.872.993.143.174.725.09 412MahdiTeimouriandArjunK.Gupta Table3:Estimatedparametersfordataset1 MethodEstimatedparametersADdistance MLE =1.171=13.550=10.1000.301 Newestimator =1.130=13.294=10.1980.378 Table4:Estimatedparametersfordataset2 MethodEstimatedparametersADdistance MLE =1:217=2:057=�0:0080.432 Newestimator =1:227=2:072=�0:0200.408 TheAnderson-Darlingstatistic(AD)isgiveninlastcolumnofeachtable.Inthesenseofthismeasure,theMLEworksbetterthanthenewestimatorfordataset1,whilethereverseisconcludedfordataset2fromTable4.4.ConclusionInthisworkanewestimatorforshapeparameter,asthemainparameter,ofthethree-parameterWeibullfamilyisproposed.Havingpropertiessuchas:closedformexpression,simplicity,asymptoticallyunbiasedness,highdegreeofperformanceandindependenceofscaleandlocationparametersmadeitasagoodcompetitorformaximumlikelihoodestimatoroftheshapeparameterwhichiscomputednumerically.Simulationsshowthatthenewestimatorworkscom-parablewiththemaximumlikelihoodestimator,inthesenseof(3.4),evenforsmallsamplesize.Tworealexamplesalsoverifythatthenewestimatorperformsverygood.ReferencesBartolucci,A.A.,Singh,K.P.,Bartolucci,A.D.andBae,S.(1999).Applyingmedicalsurvivaldatatoestimatethethree-parameterWeibulldistributionbythemethodofprobability-weightedmoments.MathematicsandCom-putersinSimulation48,385-392.Cohen,A.C.,Whitten,B.J.andDing,Y.(1984).ModiedmomentestimationforthethreeparameterWeibulldistribution.JournalofQualityTechnology16,159-167.Cramer,H.(1964).MathematicalMethodsofStatistics.PrincetonUniversityPress,Princeton. 414MahdiTeimouriandArjunK.Gupta MahdiTeimouri1-DepartmentofStatisticsFacultyofMathematics&ComputerScienceAmirkabirUniversityofTechnology424,HafezAve.,Tehran15914,Iranteimouri@aut.ac.ir2-DepartmentofStatisticsFacultyofScienceGonbadKavousUniversityShahidFallahi,163,GonbadKavous,Iranmahdiba 2001@yahoo.comArjunK.GuptaDepartmentofMathematics&StatisticsBowlingGreenStateUniversityBowlingGreen,OH43403-0221,USAgupta@bgsu.edu