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1、Chapter8:Estimation8.1LetB=B(θˆ).Then,2MSE(θˆ)=E[](θˆ−θ)2=E[(θˆ−E(θˆ)+B)2]=E⎡(θˆ−E(θˆ))⎤+E(B2)+2B×E[]θˆ−E(θˆ)⎢⎣⎥⎦2=V(θˆ)+B.8.2a.TheestimatorθˆisunbiasedifE(θˆ)=θ.Thus,B(θˆ)=0.b.E(θˆ)=θ+5.8.3a.UsingDefinition8.3,B(θˆ)=aθ+b–θ=(a–1)θ+b.*b.Letθˆ=(θˆ−b)/a.8.4a.Theyareequal.b.MSE(θˆ)>V(θˆ).
2、**28.5a.NotethatE(θˆ)=θandV(θˆ)=V[(θˆ−b)/a]=V(θˆ)/a.Then,**2MSE(θˆ)=V(θˆ)=V(θˆ)/a.2b.NotethatMSE(θˆ)=V(θˆ)+B(θˆ)=V(θˆ)+[(a−1)θ+b].Asufficientlylargevalueof*awillforceMSE(θˆ)MSE(θˆ).Example:a=.5,b=0.8.6a.E(θˆ)=aE(θˆ)+(1−a)E(θˆ
3、)=aθ+(1−a)θ=θ.312b.V(θˆ)=a2V(θˆ)+(1−a)2V(θˆ)=a2σ2+(1−a)σ2,sinceitwasassumedthatθˆand312121θˆareindependent.TominimizeV(θˆ),wecantakethefirstderivative(with23respecttoa),setitequaltozero,tofind2σ2a=.22σ+σ12(Oneshouldverifythatthesecondderivativetestshowsthatthisisindeedaminimum.)8.7Fol
4、lowingEx.8.6butwiththeconditionthatθˆandθˆarenotindependent,wefind12222V(θˆ)=aσ+(1−a)σ+2a(1−a)c.312Usingthesamemethodw/derivatives,theminimumisfoundtobe2σ−c2a=.22σ+σ−2c12158Chapter8:Estimation159Instructor’sSolutionsManual8.8a.Notethatθˆ1,θˆ2,θˆ3andθˆ5aresimplelinearcombinationsofY1,Y
5、2,andY3.So,itiseasilyshownthatallfouroftheseestimatorsareunbiased.FromEx.6.81itwasshownthatθˆhasanexponentialdistributionwithmeanθ/3,sothisestimatorisbiased.4b.ItiseasilyshownthatV(θˆ)=θ2,V(θˆ)=θ2/2,V(θˆ)=5θ2/9,andV(θˆ)=θ2/9,so1235theestimatorθˆisunbiasedandhasthesmallestvariance.58.9
6、Thedensityisintheformoftheexponentialwithmeanθ+1.WeknowthatYisunbiasedforthemeanθ+1,soanunbiasedestimatorforθissimplyY–1.8.10a.ForthePoissondistribution,E(Y)=λandsofortherandomsample,E(Y)=λ.Thus,theestimatorλˆ=Yisunbiased.2222b.TheresultfollowsfromE(Y)=λandE(Y)=V(Y)+λ=2λ,soE(C)=4λ+λ.2
7、2222c.SinceE(Y)=λandE(Y)=V(Y)+[E(Y)]=λ/n+λ=λ(1+1/n).Then,we2canconstructanunbiasedestimatorθˆ=Y+Y(4−1/n).8.11Thethirdcentralmomentisdefinedas3332E[(Y−μ)]=E[(Y−3)]=E(Y)−9E(Y)+54.Usingtheunbiasedestimatesθˆandθˆ,itcaneasilybeshownthatθˆ–9θˆ+54isan2332unbiasedestimator.8.12a.Fortheunifor
8、mdist
