資源描述：
《Mathematical Statistics with Applications, 7 edition ISM_Chapter08F》由會(huì)員上傳分享，免費(fèi)在線閱讀，更多相關(guān)內(nèi)容在學(xué)術(shù)論文-天天文庫。
1、Chapter8:Estimation8.1LetB=B(θ?).Then,2MSE(θ?)=E[](θ??θ)2=E[(θ??E(θ?)+B)2]=E?(θ??E(θ?))?+E(B2)+2B×E[]θ??E(θ?)????2=V(θ?)+B.8.2a.Theestimatorθ?isunbiasedifE(θ?)=θ.Thus,B(θ?)=0.b.E(θ?)=θ+5.8.3a.UsingDefinition8.3,B(θ?)=aθ+b–θ=(a–1)θ+b.*b.Letθ?=(θ??b)/a.8.4a.Theyareequal.b.MSE(θ?)>V(θ?).
2、**28.5a.NotethatE(θ?)=θandV(θ?)=V[(θ??b)/a]=V(θ?)/a.Then,**2MSE(θ?)=V(θ?)=V(θ?)/a.2b.NotethatMSE(θ?)=V(θ?)+B(θ?)=V(θ?)+[(a?1)θ+b].Asufficientlylargevalueof*awillforceMSE(θ?)MSE(θ?).Example:a=.5,b=0.8.6a.E(θ?)=aE(θ?)+(1?a)E(θ?
3、)=aθ+(1?a)θ=θ.312b.V(θ?)=a2V(θ?)+(1?a)2V(θ?)=a2σ2+(1?a)σ2,sinceitwasassumedthatθ?and312121θ?areindependent.TominimizeV(θ?),wecantakethefirstderivative(with23respecttoa),setitequaltozero,tofind2σ2a=.22σ+σ12(Oneshouldverifythatthesecondderivativetestshowsthatthisisindeedaminimum.)8.7Fol
4、lowingEx.8.6butwiththeconditionthatθ?andθ?arenotindependent,wefind12222V(θ?)=aσ+(1?a)σ+2a(1?a)c.312Usingthesamemethodw/derivatives,theminimumisfoundtobe2σ?c2a=.22σ+σ?2c12158Chapter8:Estimation159Instructor’sSolutionsManual8.8a.Notethatθ?1,θ?2,θ?3andθ?5aresimplelinearcombinationsofY1,Y
5、2,andY3.So,itiseasilyshownthatallfouroftheseestimatorsareunbiased.FromEx.6.81itwasshownthatθ?hasanexponentialdistributionwithmeanθ/3,sothisestimatorisbiased.4b.ItiseasilyshownthatV(θ?)=θ2,V(θ?)=θ2/2,V(θ?)=5θ2/9,andV(θ?)=θ2/9,so1235theestimatorθ?isunbiasedandhasthesmallestvariance.58.9
6、Thedensityisintheformoftheexponentialwithmeanθ+1.WeknowthatYisunbiasedforthemeanθ+1,soanunbiasedestimatorforθissimplyY–1.8.10a.ForthePoissondistribution,E(Y)=λandsofortherandomsample,E(Y)=λ.Thus,theestimatorλ?=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,we2canconstructanunbiasedestimatorθ?=Y+Y(4?1/n).8.11Thethirdcentralmomentisdefinedas3332E[(Y?μ)]=E[(Y?3)]=E(Y)?9E(Y)+54.Usingtheunbiasedestimatesθ?andθ?,itcaneasilybeshownthatθ?–9θ?+54isan2332unbiasedestimator.8.12a.Fortheunifor
8、mdist
