I型DAGUM分布形状参数的Bayes正态和T-K逼近(IJMSC-V4-N3-2)

I.J.MathematicalSciencesandComputing,2018,3,13-22PublishedOnlineJuly2018inMECS()DOI:10.5815/ijmsc.2018.03.02Availableonlineat*a,b,*DepartmentofStatistics,UniversityofKashmir,Srinagar,190006,IndiaReceived:25February2018;Accepted:15March2018;Published:08July2018AbstractDagumdistributionisastatisticaldistributionusedcloselyforfittingincomeandwealthdistributions.Thisdistributionhaswideapplicationinfieldslikereliabilitytheorysurvivalanalysis,actuarialsciences,andmeteorologicaldata.Inthisarticle,weobtainedBayesestimatorsfortheshapeparameterofDagumdistributionusingapproximationtechniqueslikenormalandT-Kapproximations.Moreoverdifferentinformativepriorshavebeenconsideredandasimulationstudyandthreerealdatasetshavebeenconsideredtostudytheefficiencyofobtainedresults.IndexTerms:Dagumdistribution,PriorDistribution,BayesianStatisticsNormalapproximation,T-Kapproximation.©2018PublishedbyMECSPublisher.Selectionand/orpeerreviewunderresponsibilityoftheResearchAssociationofModernEducationandComputerScience1.IntroductionCamiloDagum[6,7]gavethree-parametertypeIandfour-parametertypeIIandTypeIIIdistributionsforfittingincomeandwealthdistributions.However,theDagumtypeIdistributionhasreceivedmoreattentionbecausethedistributionhasmonotonicallydecreasing,upside-downbathtub,bathtubandthenupside-downbathtubhazardratefordifferentvaluesofparameterswhichledseveralauthorstostudythedistributionindifferentfieldsDommaetal.[9,11],Benjaminetal.[4].Monroyetal.[16]useditformodelingtroposphericOzonelevelsandAlwanetal.[2]workedwiththeDagumdistributionforassessingthereliabilityofanelectricalsystemandfordescribingdiameterinteakstandssubjectedtothinningatdifferentages.Differentproperties,characteristicsandparameterestimationofDagumdistributionwerestudiedbyKleiberandKotz[15],Kleiber[14],Dommaetal.[8,10],Khan[13].Brodericketal.[5]derivedanewclassofgeneralizedDagumdistributionandstudieditsapplicationstoincomeandlifetimedatatoillustratetheusefulnessofthemodel.AalaAhmed[1]proposedtheestimatesandasymptoticdistributionofDagumdistribution.Tahiretal.*Correspondingauthor.E-mailaddress:14BayesianNormalandT-KApproximationsforShapeParameterofType-IDagumDistribution[23]definedanewlifetimemodelWeibull-Dagumdistributionstudieditsstructuralpropertiesandillustrateditspotentialitybymeansofsimulationstudyandreallifeapplications.TheprobabilitydensityfunctionofDagumdistribution0,,;)1(),,:(11yyyf(1)whereandareshapeparametersandisthescaleparameter.Thelikelihoodfunctionof(1.1)isgivenas))1(ln)1(ln)1(exp(),,:(11iniininnnyyyL(2)TheaimofourpresentstudyistoobtaintheBayesestimatesoftheshapeparameterofType-IDagumdistributionusingnormalapproximationandT-Kapproximationtechniquesunderdifferentinformativepriors.2.BayesianApproximationTechniquesofPosteriorModesBayesianinferenceprovidesarationalmethodforupdatingbeliefsinlightofnewinformation.Bayesiananalysisisbasedonthepremisethatalluncertaintyshouldbemodeledusingprobabilitiesandthatstatisticalinferenceshouldbelogicalconclusionsbasedonthelawsofprobability.Itmaybenotedthatposteriordistributiontakesaratiothatinvolvesintegrationinthedenominatorandcannotbereducedtoclosedform.HencetheevaluationoftheposteriorexpectationforobtainingtheBayesestimatorswillbetedious.Thus,weproposetheuseofBayesianapproximationtechniquesforobtainingBayesestimates.Iftheposteriordistributionx|isunimodalandroughlysymmetric,itisconvenienttoapproximateitbyanormaldistributioncenteredatthemode,yieldingtheapproximation1ˆ,ˆˆ~|INx,where22|logˆyPI(3)Ifthemode,ˆisintheinteriorparameterspace,thenIispositive;ifˆisavectorparameter,thenIisamatrix.TierneyandKadane[24]gaveLaplacemethodtoevaluate)|)((xhEas)}ˆ()ˆ(exp{)|)((hnhnxhE,where)(ln)|(ln)ˆ(**hxnh,12)ˆ(ˆhn,1**2*)ˆ(ˆhnRecentlySultanetal.[19,20,21,22]obtainedtheBayesestimatesforTopp-LeoneDistribution,Kumaraswamydistribution,generalizedpowerfunctiondistribution,andgeneralizedgammadistributionusingBayesianapproximationtechniques.Naqashetal.[17]proposedaBayesiananalysisofDagumdistributionforthecompletesampleunderdifferentlossfunctionandpriors.BayesianNormalandT-KApproximationsforShapeParameterofType-IDagumDistribution153.BayesianNormalApproximationforShapeParameterofType-IDagumDistributionInthissection,theestimatesofshapeparameterunderdifferentpriorsareobtainedusingnormalapproximationtechnique.ThenormalapproximationsforType-IDagumdistributionunderMukherjee-IslamPrior)1(1)(bg,GammaPrior0,;)(11111dcegdcandInverseLevyPrior0;)(225.05.0aegaareobtainedas:PosteriordensityofundertheMukherjee-IslamPrioris))1ln()1(ln)1(exp()|(1111iniinibnyyy(4)Fromwhich)1ln(1)|(lnˆ11iniybnyand2111)]1ln([)1()ˆ(iniybnITherefore21111)]1ln([)1(,)]1ln([)1(~)|(iniiniybnybnNyTheposteriordensityofun