hw5_stat210a

# hw5_stat210a - UC Berkeley Department of Statistics STAT...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5 Fall 2006 Issued: Thursday, September 28, 2006 Due: Thursday, October 5, 2006 A useful de±nition for this problem set: Defnition: An equalizer procedure is a rule with constant risk (i.e., R ( θ, δ ) = c for all θ Θ). Problem 5.1 Let Θ = (0 , + ) and A = [0 , ), and suppose that X Poi( θ ). Consider the loss function L ( θ, a ) = ( θ - a ) 2 . (a) Show that the estimator δ ( X ) = X is an equalizer rule. (b) Show that δ is generalized Bayes with respect to the improper prior that is uniform on (0 , ). (c) Find Bayes estimators with respect to the family of Gamma( a, b ) priors. Problem 5.2 For θ (0 , 1), let X Bin( n, θ ), and consider the weighted quadratic loss function L ( θ, a ) = ( θ - a ) 2 θ (1 - θ ) . (Note that this loss function penalizes more severely for extreme values of θ near 0 or 1.) (a) Show that for this loss function, δ

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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 Berkeley.

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hw5_stat210a - UC Berkeley Department of Statistics STAT...

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