{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

stat210a_2007_hw6_solutions

stat210a_2007_hw6_solutions - UC Berkeley Department of...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

Page1 / 5

stat210a_2007_hw6_solutions - UC Berkeley Department of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online