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1、Ch7.PointestimationIntroDefinitionAfunctionofrandomvariablesX1,···,XniscalledStatistic.DefinitionTheprobabilitydistributionofastatisticTiscalledthesamplingdistributionofT.iid2BExample:X1,···,Xn∼N(µ,σ).LetT=X¯,thensamplingdistributionofTisN(,).IfT=S2thensam
2、plingdistributionofS2is(n−1)S2/σ2∼.Ch7.PointestimationIntroDefinitionAnestimatorisafunctionofrandomvariablesX1,···,Xn,T=W(X1,···,Xn).CNote:1.Estimatorisactuallyastatistic.2.Estimatorisalsorandom.3.AnestimateisafunctionofrealizedvaluesofX1=x1,···,Xn=xn.t=W
3、(x1,···,xn)−1PnBExample:T=X¯,Fˆn(x0)=ni=1I[Xi≤x0],T=(X,S¯2).Ch7.PointestimationMethods:MMEHowtoestimatetheparametricfunctionτ(θ)usingtherandomsampleX1,···,Xn?:MME,MLE,BEsooniidLetX1,···,Xn∼f(x
4、θ),θ=(θ1,···,θk)∈Θ.Definition jthPopulationmoment:µ(θ)=EXj
5、j
6、
7、Xn
8、th1jjSamplemoment:mj=Xi
9、ni=1CNote:1.µjisafunctionofθ=(θ1,···,θk).hPi2.E[m]=En−1nXj=µ.ji=1ijCh7.PointestimationMethods:MMEDefinitionMMEofθ=(θ1,···,θk),denotedbyθ˜=(θ˜1,···,θ˜k),isdefinedasasolutionofthesystemofequationsmj=µj(θ1,···,θk),j=1,···.liid22BExa
10、mple:X1,···,Xn∼N(µ,σ).FindMMEofθ=(µ,σ).Ch7.PointestimationMethods:MMEiidBExample:X1,···,Xn∼Gamma(α,β).FindMMEofθ=(α,β).Let***Ch7.PointestimationMethods:MMECNote:1.MMequationsmayhavemultiplesolutionsornosolution.Thesolutionmayfalloutsideoftheparameterspac
11、e.2.MMEmaynotbeapplicableifthepopulationmomentsdonotexistsuchasCauchydistribution.3.Onemaynotsuccessfulconsideringthefirstk–moments.iidBExample:X1,···,Xn∼f(x
12、β),where1−
13、x
14、/βf(x
15、β)=e.2βCh7.PointestimationMethods:MLEGiventhesampleobservation(measurement,dat
16、a),chooseastheestimatorofthepopulationparameterthevaluethatmakestheobservedsampleresultthemostlikely,i.e.maximizethelikelihoodBLetXbeasingleobservationfromadiscretedistributiontakingthevalues{0,1,2}.Thedatacomesfromoneoftwodistributions:Ch7.Pointestimati
17、onMethods:MLEDefinitionLetf(x1,···,xn
18、θ)bethejointpdf/pmfofX1,···,Xn.Forafixedx1,···,xn,L(θ)=f(x1,···,xn
19、θ)asafunctionofθ,iscalledthelikelihoodfunction.ln[L(θ)]iscalledtheloglikelihoodfunction.Withdiscreterandomvariable,L(θ)
