7-18Solutions Manual for Statistical InferenceTo calculate the Cram´er-Rao lower bound, we haveEθ∂2logf(X|θ)∂θ2=Eθ∂2∂θ2log1√2πe-(X-θ)2/2=Eθ∂2∂θ2log(2π)-1/2-12(X-θ)2=Eθ∂∂θ(X-θ)=-1,andτ(θ) =θ2, [τ(θ)]2= (2θ)2= 4θ2so the Cram´er-Rao Lower Bound for estimatingθ2is[τ(θ)]2-nEθ(∂2∂θ2logf(X|θ))=4θ2n.Thus, the UMVUE ofθ2does not attain the Cram´er-Rao bound. (However, the ratio of thevariance and the lower bound→1 asn→ ∞.)7.45 a. Because ES2=σ2, bias(aS2) = E(aS2)-σ2= (a-1)σ2. Hence,MSE(aS2) = Var(aS2) + bias(aS2)2=a2Var(S2) + (a-1)2σ4.b. There were two typos in early printings;κ= E[X-μ]4/σ4andVar(S2) =1nκ-n-3n-1σ4.See Exercise 5.8b for the proof.c. There was a typo in early printings; under normalityκ= 3. Under normality we haveκ=E[X-μ]4σ4= EX-μσ4= EZ4,whereZ∼n(0,1). Now, using Lemma 3.6.5 with
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Ratio, early printings, cram´r-rao lower bound, n-3 n-1