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 n is proper prior and = lim n n is improper prior, in general lim n Z R ( , n ) n ( ) d 6 = Z lim n R ( , n ) n ( ) d 6 = Z R ( , ) ( ) d where n , is related Bayes estimator. Theorem Let n be a Bayes estimator w.r.t. a proper prior n , n = 1 ,...,n . If max R ( , * ) lim n inf Z R ( , n ) n ( ) d Then, * is minimax estimator. Proof Z max R ( , )- R ( , ) n ( ) d , n Z { R ( , )- R ( , n ) } n ( ) d , n Thus, Z max R ( , )- R ( , n ) n ( ) d = max R ( , )- Z R ( , n ) n ( ) d , n Therefore, min max R ( , ) lim n inf Z R ( , n ) n ( ) d max R ( , * ) 1 Let prior k Uni (- k,k ). Then, for sufficiently large k , k ( X ) = E ( | X 1 ,...,X n ) = X ( n )- 1 / 2 + X (1) + 1 / 2 2 = X (1) + X ( n ) 2 ANd because X i- i.i.d. Uni (0 , 1) E X (1) + X ( n ) 2- 2 = E ( X (1)- ) + ( X ( n )- ) 2 2 = constant max R ( , * ) = lim k inf Z R ( , k ) k ( ) d for * ( X ) = X (1) + X ( n ) 2 ....
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This note was uploaded on 10/17/2009 for the course STAT 210a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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stat210a_2007_hw6_solutions - UC Berkeley Department of...

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