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Dr. Hackney STA Solutions pg 115

Dr. Hackney STA Solutions pg 115 - 7-18 Solutions Manual...

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7-18 Solutions Manual for Statistical Inference To calculate the Cram´ er-Rao lower bound, we have E θ 2 log f ( X | θ ) ∂θ 2 = E θ 2 ∂θ 2 log 1 2 π e - ( X - θ ) 2 / 2 = E θ 2 ∂θ 2 log(2 π ) - 1 / 2 - 1 2 ( X - θ ) 2 = E θ ∂θ ( X - θ ) = - 1 , and τ ( θ ) = θ 2 , [ τ ( θ )] 2 = (2 θ ) 2 = 4 θ 2 so the Cram´ er-Rao Lower Bound for estimating θ 2 is [ τ ( θ )] 2 - n E θ ( 2 ∂θ 2 log f ( X | θ ) ) = 4 θ 2 n . Thus, the UMVUE of θ 2 does not attain the Cram´ er-Rao bound. (However, the ratio of the variance and the lower bound 1 as n → ∞ .) 7.45 a. Because E S 2 = σ 2 , bias( aS 2 ) = E( aS 2 ) - σ 2 = ( a - 1) σ 2 . Hence, MSE( aS 2 ) = Var( aS 2 ) + bias( aS 2 ) 2 = a 2 Var( S 2 ) + ( a - 1) 2 σ 4 . b. There were two typos in early printings; κ = E[ X - μ ] 4 4 and Var( S 2 ) = 1 n κ - n - 3 n - 1 σ 4 . See Exercise 5.8b for the proof. c. There was a typo in early printings; under normality κ = 3. Under normality we have κ = E[ X - μ ] 4 σ 4 = E X - μ σ 4 = E Z 4 , where Z n(0 , 1). Now, using Lemma 3.6.5 with
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