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1、Chapter3UnbiasedEstimationUnbiasedorasymptoticallyunbiasedestimationplaysanimportantroleinpointestimationtheory.Unbiasednessofpointestimatorsisde?nedin§2.3.2.Inthischapter,wediscussindetailhowtoderiveunbiasedesti-matorsand,moreimportantly,howto?ndthebestunbiasedestimatorsin
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∈Pand?beareal-va
4、luedparameterrelatedtoP.RecallthatanestimatorT(X)of?isunbiasedifandonlyifE[T(X)]=?foranyP∈P.Ifthereexistsanunbiasedestimatorof?,then?iscalledanestimableparameter.De?nition3.1.AnunbiasedestimatorT(X)of?iscalledtheuni-formlyminimumvarianceunbiasedestimator(UMVUE)ifandonlyifVa
5、r(T(X))≤Var(U(X))foranyP∈PandanyotherunbiasedestimatorU(X)of?.Sincethemseofanyunbiasedestimatorisitsvariance,aUMVUEis?-optimalinmsewith?beingtheclassofallunbiasedestimators.One1611623.UnbiasedEstimationcansimilarlyde?netheuniformlyminimumriskunbiasedestimatorinsta-tisticald
6、ecisiontheorywhenweuseanarbitrarylossinsteadofthesquarederrorlossthatcorrespondstothemse.3.1.1Su?cientandcompletestatisticsThederivationofaUMVUEisrelativelysimpleifthereexistsasu?cientandcompletestatisticforP∈P.Theorem3.1(Lehmann-Sche?′etheorem).Supposethatthereexistsasu?ci
7、entandcompletestatisticT(X)forP∈P.If?isestimable,thenthereisauniqueunbiasedestimatorof?thatisoftheformh(T)withaBorelfunctionh.(Twoestimatorsthatareequala.s.Paretreatedasoneestimator.)Furthermore,h(T)istheuniqueUMVUEof?.ThistheoremisaconsequenceofTheorem2.5(ii)(Rao-Blackwell
8、the-orem).Onecaneasilyextendthistheoremtothecaseoftheuniformlyminimumriskunbiasede
