基于记录值的指数分布参数的极大极小估计(IJITCS-V6-N3-6)

I.J.InformationTechnologyandComputerScience,2014,03,47-53PublishedOnlineFebruary2014inMECS()DOI:10.5815/ijitcs.2014.03.06Copyright©2014MECSI.J.InformationTechnologyandComputerScience,2014,03,47-53MinimaxEstimationoftheParameterofExponentialDistributionbasedonRecordValuesLanpingLiDepartmentofBasicSubjects,HunanUniversityofFinanceandEconomics,Changsha410205,P.R.ChinaE-mail:[e-mail]—Bayesestimatorsoftheparameterofexponentialdistributionareobtainedwithnon-informativequasi-priordistributionbasedonrecordvaluesunderthreelossfunctions.Thesefunctionsareweightedsquarederrorloss,squarelogerrorlossandentropylossfunctions.FinallytheminimaxestimatorsoftheparameterareobtainedbyusingLehmann’stheorem.Comparisonsintermsofriskswiththeestimatorsofparameterunderthreelossfunctionsarealsostudied.IndexTerms—BayesEstimator,MinimaxEstimator,SquaredLogErrorLoss,EntropyLoss,RecordValueI.IntroductionRecordvaluesandtheassociatedstatisticsareaofinterestandimportantinmanyreallifeapplicationsinvolvingdatarelatingtometeorology,sport,economicsandlifetesting.Set1},,,,max{21nXXXYnn,wesaythatjXisanupperrecordanddenotedby)(jUXif1,1jYYjj.ThedetailaboutrecordvaluescanreferArnoldetal.(1998),Raqab(2002),Jaheen(2004)andAhmadietal.(2005)andreferencestherein.Exponentialdistributionisoneofthemostcommonlyusedmodelsinlife-testingandreliabilitystudies.Inferentialissuesconcerningtheexponentialdistribution,andapplicationsinthecontextoflife-testingandreliability,havebeenextensivelydiscussedbymanyscholars.AgreatdealofresearchhasbeendoneonestimatingtheparametersoftheexponentialdistributionusingbothclassicalandBayesiantechniques.SeeforexampleBain(1978),Chandrasekaretal.(2002),Jaheen(2004),andAhmadietal.(2005)andreferencestherein.Thispaperisdevotedtotheminimaxestimationproblemoftheunknownscaleparameterintheexponentialdistributionwithprobabilitydensityfunction(pdf)0,0,x)exp(-);(xxf(1)andcumulativedistributionfunction(cdf)0,0,x)exp(-1);(xxF(2)where(;)fxdenotestheconditionalpdfofrandomvariable(r.v.)Xgiven.Thispaperwilldiscusstheminimaxestimationoftheparameterofexponentialdistributionbasedonrecordvalues.II.Preliminaries2.1MaximumLikelihoodEstimationLet,,21XXbeasequenceofindependentandidenticallydistributed(iid)randomvariableswithcdf);(xFandpdf);(xf.Inthefollowingdiscussion,wealwayssupposethatweobservenupperrecordvaluesnnUUUxXxXxX)(2)2(1)1(,,,drawnfromtheexponentialmodelwithpdfgivenby(1).Thejointdistributionof)()2()1(,,,nUUUXXXisgiven(seeArnoldetal.(1998))by11,2,,11(;)(;)(;),0nnniinfxfxhxxx(3)Where);(1);();(),,,,(21iiinxFxfxhxxxxSincethemarginalpdfof)(nUXisgiven(seeArnoldetal.(1998))by48MinimaxEstimationoftheParameterofExponentialDistributionbasedonRecordValuesCopyright©2014MECSI.J.InformationTechnologyandComputerScience,2014,03,47-5311[ln(1(;)](;)(;)(1)!1,0()nnnnnnxnnnFxfxfxnxexn(4)Thus),(~)(nXnU,and/)(nEXnUThelikelihoodfunctionbasednupperrecordvaluesisgivenby)exp();()|(,,2,1nnnxxfxl(5)andthelog-likelihoodfunctionmaybewrittenasnxnxlxLln)|(ln)|((6)Upondifferentiating(6)withrespecttoandequatingeachresultstozero,theMLEofisgivenby0)|(lnnxndxLdThen，theMLEofis)(ˆnUMLEXn(7)2.2LossFunctionInstatisticaldecisiontheoryandBayesiananalysis,lossfunctionplaysanimportantroleinitandthemostcommonlossaresymmetriclossfunction,especiallysquarederrorlossfunctionareconsideredmost.Undersquaredlossfunction,itistobethoughttheoverestimationandunderestimationhavethesameestimatedrisks.However,inmanypracticalpracticalproblems,overestimationandunderestimationwillhavedifferentconsequences.Toovercomethisdifficulty,Varian(1975)andZellner(1986)proposedanasymmetriclossfunctionknownastheLINEXlossfunction,Podderetal.(2004)proposedanewasymmetriclossfunctionforscaleparameterestimation.SeealsoKiapouraandNematollahib(2011),Mahmoodi,andFarsipour(2006).Thislossfunctioniscalledsquaredlogerrorloss(SLE)is2)ln(ln),(L(8)Whichisbalancedand),(Las0or.Thislossisnotalwaysconvex,itisconvexforeandconcaveotherwise,butitsriskfunctionhasminimumw.r.t.)]|(lnexp[ˆXESL.Inmanypracticalsituations,itappearstobemorerealistictoexpressthelossintermsoftheratio/ˆ.Inthiscase,Deyetal.(1987)pointedoutthatausefulasymmetriclossfunctionisentropylossfunction:ˆ(,)ln1L(9)Whoseminimumoccursat.Also,thislossfunctionhasbeenusedinSinghetal.(2011),LiandRen(2012).TheBayesestimatorundertheentropylossisdenotedbyBEˆ,givenby11ˆ[(|)]BEEX.(10)III.BayesEstimationInthissection,weestimatebyconsideringweightedsquareerrorloss,squaredlogerrorlossandentropylossfunctions.Wefurtherassumethatsomepriorknowledgeabouttheparameterisavailabletotheinvestigationfrompastexperiencewiththeexponentialmodel.Thepriorknowledgecanoftenbesummarizedintermsoftheso-calledpriordensitiesonparameterspaceof.Inthefollowingdiscussion,weassumethefollowingJeffrey’snon-informativequasi-priordensitydefinedas,1(),0d(11)Hence,0dleadstoadiffusepriorand1dtoanon-informativeprior.Combingthelikelihoodfunction(5)andth