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hw5_stat210a_solutions

hw5_stat210a_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 5 Fall 2006 Issued: Thursday, September 14, 2006 Due: Thursday, September 21, 2006 Graded exercises Problem 5.1 a) We want to show that R ( θ,δ ) = E θ ( ( X- θ ) θ ) does not depend on θ . To do that write: E θ ( ( X- θ ) 2 θ ) = E θ ( X- θ ) 2 θ = θ θ = 1 where the second equality follows from the fact that the variance of a Poi( θ ) random variable has variance θ . b) From item c below, we have that for the Gamma( a,b ) prior a Bayes estimator is: δ ( X ) = a + ∑ i X i- 1 n + b By constructing a sequence of priors with b k ↓ 0 and a k → 1, we have: δ k ( X ) = a k- 1 + ∑ i X i b k + n Hence: δ k ( X ) → ∑ i X i n and the sequence of priors approaches the “uniform” distribution on R + . c) We have that: π ( θ ) = exp[( a- 1)log( θ )- bθ + a log( b )- log Γ( a )] and p ( x | θ ) = exp "- nθ + X i x i log( θ ) # 1 Q i x i ! So: p ( θ | x ) ∝ exp " ( a + X i X i- 1)log( θ )- ( b + n ) θ # 1 which corresponds to a Γ( a + ∑ i X i ,b + n ) family (could also get this from Γ being conjugate prior to Poisson).conjugate prior to Poisson)....
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hw5_stat210a_solutions - UC Berkeley Department of...

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