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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 ( θ, δ ) = c for all θ ∈ Θ). 6.1. Consider the Bayesian model in which Θ has distribution Λ, and conditioned on Θ = θ , the random variable X has distribution P θ . Sup- pose that we are interested in estimating g ( θ ) under quadratic loss. Prove that no unbiased estimator δ ( X ) of g ( θ ) can be a Bayesian estimator unless E [( δ ( X )- g (Θ)) 2 ] = 0, where the expectation is over the joint distribution of ( X, Θ). Hint: Derive two different expresssions for E [ δ ( X ) g (Θ)] 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( θ ), and suppose that we place a beta B ( a, b ) prior on θ . (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
- Statistics, Probability theory, Discrete probability distribution, θ, Bayes estimator