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stat210a_2007_hw6_solutions

# stat210a_2007_hw6_solutions - UC Berkeley Department of...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 6- Solutions Fall 2007 Issued: Thursday, October 4 Due: Thursday, October 11 Problem 6.1 Note : You can NOT apply the Theorem 11.1 with improper prior. Please check the proof carefully. Note that if λ n is proper prior and λ = lim n →∞ λ n is improper prior, in general lim n →∞ Z R ( θ,δ n ) λ n ( θ ) dθ 6 = Z lim n →∞ R ( θ,δ n ) λ n ( θ ) dθ 6 = Z R ( θ,δ ) λ ( θ ) dθ where δ n ,δ is related Bayes estimator. Theorem Let δ n be a Bayes estimator w.r.t. a proper prior λ n , n = 1 ,...,n . If max θ ∈ Θ R ( θ,δ * ) ≤ lim n →∞ inf Z R ( θ,δ n ) λ n ( θ ) dθ Then, δ * is minimax estimator. Proof Z ‰ max η ∈ Θ R ( η,δ )- R ( θ,δ ) λ n ( θ ) dθ ≥ ∀ δ, ∀ λ n Z { R ( θ,δ )- R ( θ,δ n ) } λ n ( θ ) dθ ≥ ∀ δ, ∀ λ n Thus, Z ‰ max η ∈ Θ R ( η,δ )- R ( θ,δ n ) λ n ( θ ) dθ = max η ∈ Θ R ( η,δ )- Z R ( θ,δ n ) λ n ( θ ) dθ ≥ ∀ δ, ∀ λ n Therefore, min δ max η ∈ Θ R ( η,δ ) ≥ lim n →∞ inf Z R ( θ,δ n ) λ n ( θ ) dθ ≥ max θ ∈ Θ R ( θ,δ * ) 1 Let prior θ k ∼ Uni (- k,k ). Then, for sufficiently large k , δ k ( X ) = E ( θ | X 1 ,...,X n ) = X ( n )- 1 / 2 + X (1) + 1 / 2 2 = X (1) + X ( n ) 2 ANd because X i- θ ∼ i.i.d. Uni (0 , 1) E X (1) + X ( n ) 2- θ ¶ 2 = E ( X (1)- θ ) + ( X ( n )- θ ) 2 ¶ 2 = constant max θ ∈ Θ R ( θ,δ * ) = lim k →∞ inf Z R ( θ,δ k ) λ k ( θ ) dθ for δ * ( X ) = X (1) + X ( n ) 2 ....
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stat210a_2007_hw6_solutions - UC Berkeley Department of...

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