逆指数分布的贝叶斯逼近技术及其工程应用(IJMSC-V4-N2-5)

I.J.MathematicalSciencesandComputing,2018,2,49-62PublishedOnlineApril2018inMECS()DOI:10.5815/ijmsc.2018.02.05Availableonlineat:06February2018;Accepted:15March2018;Published:08April2018AbstractThepresentstudyisconcernedwiththeestimationofInverseExponentialdistributionusingvariousBayesianapproximationtechniqueslikenormalapproximation,TierneyandKadane(T-K)Approximation.Differentinformativeandnon-informativepriorsareusedtoobtaintheBaye’sestimateofInverseExponentialdistributionunderdifferentapproximationtechniques.AsimulationstudyhasalsobeenconductedforcomparisonofBaye’sestimatesobtainedunderdifferentapproximationusingdifferentpriors.IndexTerms:BayesianEstimation,PriorDistribution,NormalApproximation,T-KApproximation.©2018PublishedbyMECSPublisher.Selectionand/orpeerreviewunderresponsibilityoftheResearchAssociationofModernEducationandComputerScience1.IntroductionTheInverseExponentialdistributionwasintroducedbyKellerandKamath(1982).Duetoitsinvertedbathtubfailurerate,itissignificantcompetitivemodelfortheExponentialdistribution.AcomprehensivedescriptionofthismodelisgivenbyLinetal(1989)asalifetimemodel.ConsiderarandomvariableXfollowinganexponentialdistributionandthenthevariableXY1willhaveaninvertedexponentialdistribution(IED).Thus,ifXfollowstheInverseExponentialdistributionwithascaleparameter0,thenthecumulativedensityfunction(cdf)andtheprobabilitydensityfunction(pdf)ofthemodelarerespectivelygivenby:*Correspondingauthor.E-mailaddress:[e-mail]50BayesianApproximationTechniquesofInverseExponentialDistributionwithApplicationsinEngineering2();0,0xfxexx(1)ThecdfofXis:();0,0xFxex(2)InvertedexponentialdistributionasalifedistributionmodelfromaBayesianviewpointwasconsideredbySankuDey(2007)whileGyanPrakash(2012)obtainedtheestimationoftheInvertedexponentialdistributionbyusingsymmetricandasymmetriclossfunctions.Singhetal(2015)discussedtheestimationofstressstrengthreliabilityparameterofinvertedexponentialdistribution.TheyobtainedBayesestimatorforparametersofinvertedexponentialdistributionbyusinginformativeandnon-informativepriors.TheyalsocomparedtheclassicalmethodwithBayesianmethodthroughthesimulationstudy.2.MaterialandMethodsTheBayesianparadigmisconceptuallysimpleandprobabilisticallyelegant.Sometimesposteriordistributionisexpressibleintermsofcomplicatedanalyticalfunctionandrequiresintensivecalculationbecauseofitsnumericalimplementations.Itisthereforeusefultostudyapproximateandlargesamplebehaviorofposteriordistribution.Thus,ourpresentstudyfocusestoobtaintheestimatesoftheparameterofinvertedexponentialdistributionusingtwoBayesianapproximationtechniquesi.e.normalapproximationandT-Kapproximation.3.NormalApproximationThebasicresultofthelargesampleBayesianinferenceisthattheposteriordistributionoftheparameterapproachesanormaldistribution.Iftheposteriordistributionxp|isunimodalandroughlysymmetric,itisconvenienttoapproximateitbyanormaldistributioncenteredatthemode;thatislogarithmoftheposteriorisapproximatedbyaquadraticfunction,yieldingtheapproximation1)]ˆ([,ˆ~|INxp,where22)|(log)ˆ(xPI(3)Ifthemode,ˆisintheinteriorparameterspace,then)(Iispositive;ifˆisavectorparameter,then)(Iisamatrix.SomegoodsourcesonthetopicisprovidedbyAhmadet.al(2007,2011)discussedBayesiananalysisofexponentialdistributionandgammadistributionusingnormalandLaplaceapproximations.Sultanetal.(2015a,2015b)obtainedtheBaye’sestimatesunderdifferentinformativeandnon-informativepriorsofshapeparameterofTopp-LeoneandKumaraswamyDistributionusingBayesianapproximationtechniques.BayesianApproximationTechniquesofInverseExponentialDistributionwithApplicationsinEngineering51InourstudythenormalapproximationsofInverseExponentialdistributionunderdifferentpriorsistobeobtainedasunder:Thelikelihoodfunctionof(1)forasampleofsizenisgivenas.)|(112niiixniinexxL(4)UnderQuasiprior0,1dgd,theposteriordistributionforisasTdnexP|,whereniiixT1.(5)ThefirstderivativeisTdnxP|log,fromwhichtheposteriormodeisobtainedasTdn)(ˆ.Toconstructtheapproximation,weneedthesecondderivativesofthelog-posteriordensity,TdnxPlogconstantlog|log.(6)Thesecondderivativeofthelog-posteriordensityis222)(|logdnxP,andhence,negativeofHessianis21222)()]ˆ([)(|log)ˆ(TdnIdnTxPI.Thus,theposteriordistributioncanbeapproximatedas2)(,)(~|TdnTdnNxp.(7)UnderextensionofJeffrey’spriorRcgc12,11,theposteriordistributionforisas52BayesianApproximationTechniquesofInverseExponentialDistributionwithApplicationsinEngineeringTcnexP12|,whereniiixT1(8)TcnxPlog2constantlog|log1.(9)ThefirstderivativeisTcnxP12|log,fromwhichtheposteriormodeisobtainedasTcn)2(ˆ1.Thesecondderivativeofthelog-posteriordensityis2122)2(|logcnxP,andhence,negativeofHessi