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1、CHAPTER12Appendix:BackgroundMaterialsInthefollowingsections,weprovidesomebackgroundmaterialsforthestan-dardmethodsthathavebeenrepeatedlyusedthroughoutthebook,includingthelikelihoodmethodsandMCMCmethods.Wewillfocusonessentialideasandresults,withoutgoingtoomuchdetails.Moredetaileddiscussionsofthesetop
2、icscanbefoundinmanybooks,whicharelistedinthecorrespondingsec-tions.12.1LikelihoodMethodsLikelihoodmethodsarewidelyusedinstatisticalinference,duetogeneralap-plicabilityoflikelihoodmethodsandattractiveasymptoticpropertiesofMLEssuchasasymptoticmostefciencyandasymptoticnormality.Moreover,thelikelihoodp
3、rinciplesaysthatlikelihoodfunctionscontainalloftheinforma-tioninthedataaboutunknownparametersintheassumedmodels.Maximumlikelihoodestimationisoftenviewedasthe“goldstandard”ofestimationpro-cedures.LikelihoodfunctionsalsoplayanintegralroleinBayesianinference.Inthefollowing,weprovideabriefoverviewoflike
4、lihoodmethods.Foralikelihoodmethod,oncethelikelihoodfortheobserveddataisspeci-edbasedontheassumeddistributions,theMLEsofunknownparametersintheassumeddistributionscanbeobtainedbymaximizingthelikelihoodusingstandardoptimizationproceduresortheEMalgorithms.TheresultingMLEswillbeasymptoticallyconsistent
5、,mostefcient(inthesenseofattainingtheCramer-RaolowerboundforthevariancesoftheMLEs),andnormallydis-tributed,ifsomecommonregularityconditionshold.Inotherwords,whenthesamplesizeislarge,theMLEisapproximatelyoptimaliftheassumeddis-tributionsandsomeregularityconditionshold.Inmanyproblems,thesamplesizesdo
6、nothavetobeverylargeinorderfortheMLEstoperformwell,andtheregularityconditionsareoftensatised.Violationsoftheregularitycondi-tionsmayarise,forexample,whentheparametersareontheboundaryofthe375376MIXEDEFFECTSMODELSFORCOMPLEXDATAparameterspace.Therefore,likelihoodmethodsareconceptuallystraightfor-ward.
7、Inpractice,difcultiesoftenlieincomputationsincetheobserved-datalikelihoodscanbehighlyintractableforsomecomplexproblems.TheasymptoticnormalityofMLEscanbeusedfor(approximate)inferenceinpracticewherethe
