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Unformatted text preview: STAT210A HW06 Due: Tuesday, October 13, 2009 A useful definition for this problem set: Definition: An equalizer procedure is a rule with constant risk (i.e., R ( , ) = c for all ). 6.1. Consider the Bayesian model in which has distribution , and conditioned on = , the random variable X has distribution P . Sup- pose that we are interested in estimating g ( ) under quadratic loss. Prove that no unbiased estimator ( X ) of g ( ) can be a Bayesian estimator unless E [( ( X )- g ()) 2 ] = 0, where the expectation is over the joint distribution of ( X, ). Hint: Derive two different expresssions for E [ ( X ) g ()] by using the tower property of conditional expectation. 6.2. Suppose that we have n i.i.d. samples of the form X i Bernoulli( ), and suppose that we place a beta B ( a, b ) prior on . (a) Calculate the Bayes risk of the Bayes estimator. (See Example 9.2.2 of Keener; also discussed in lecture.) (b) Now suppose the alternative sampling model, in which we perform the...
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- Fall '08