Mathematical Statistics - Jun Shaochap3

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1、Chapter3UnbiasedEstimationUnbiasedorasymptoticallyunbiasedestimationplaysanimportantroleinpointestimationtheory.Unbiasednessofpointestimatorsisdefinedin§2.3.2.Inthischapter,wediscussindetailhowtoderiveunbiasedesti-matorsand,moreimportantly,howtofindthebestunbiasedestimatorsin

2、varioussituations.Althoughanunbiasedestimator(eventhebestunbiasedestimatorifitexists)isnotnecessarilybetterthanaslightlybiasedesti-matorintermsoftheirmse’s(seeExercise63in§2.6),unbiasedestimatorscanbeusedas“buildingblocks”fortheconstructionofbetterestimators.Furthermore,one

3、maygiveuptheexactunbiasedness,butcannotgiveupasymptoticunbiasednesssinceitisnecessaryforconsistency(see§2.5.2).Propertiesandtheconstructionofasymptoticallyunbiasedestimatorsarestudiedinthelastpartofthischapter.3.1TheUMVUELetXbeasamplefromanunknownpopulationP∈Pandϑbeareal-va

4、luedparameterrelatedtoP.RecallthatanestimatorT(X)ofϑisunbiasedifandonlyifE[T(X)]=ϑforanyP∈P.Ifthereexistsanunbiasedestimatorofϑ,thenϑiscalledanestimableparameter.Definition3.1.AnunbiasedestimatorT(X)ofϑiscalledtheuni-formlyminimumvarianceunbiasedestimator(UMVUE)ifandonlyifVa

5、r(T(X))≤Var(U(X))foranyP∈PandanyotherunbiasedestimatorU(X)ofϑ.Sincethemseofanyunbiasedestimatorisitsvariance,aUMVUEisℑ-optimalinmsewithℑbeingtheclassofallunbiasedestimators.One1611623.UnbiasedEstimationcansimilarlydefinetheuniformlyminimumriskunbiasedestimatorinsta-tisticald

6、ecisiontheorywhenweuseanarbitrarylossinsteadofthesquarederrorlossthatcorrespondstothemse.3.1.1SufficientandcompletestatisticsThederivationofaUMVUEisrelativelysimpleifthereexistsasufficientandcompletestatisticforP∈P.Theorem3.1(Lehmann-Scheff´etheorem).Supposethatthereexistsasuffici

7、entandcompletestatisticT(X)forP∈P.Ifϑisestimable,thenthereisauniqueunbiasedestimatorofϑthatisoftheformh(T)withaBorelfunctionh.(Twoestimatorsthatareequala.s.Paretreatedasoneestimator.)Furthermore,h(T)istheuniqueUMVUEofϑ.ThistheoremisaconsequenceofTheorem2.5(ii)(Rao-Blackwell

8、the-orem).Onecaneasilyextendthistheoremtothecaseoftheuniformlyminimumriskunbiasede