stat210a_2007_hw5_solutions

stat210a_2007_hw5_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- Solutions Fall 2007 Issued: Thursday, September 27 Due: Thursday, October 4 Problem 5.1 (a) For the Gamma density, we have: p ( x | θ ) ∝ exp • λ log( x )- px + log p λ Γ( λ ) ¶ I ( p ≥ 0) I ( λ ≥ 0) ‚ And hence, a conjugate family is an exponential family on ( λ,p ) with natural statistics λ,p, log ‡ p λ Γ( λ ) · , that is: π ( λ,p ) = exp • t 1 λ + t 2 p + t 3 log p λ Γ( λ ) ¶- ω ( t ) ‚ I ( p ≥ 0) I ( λ ≥ 0) Ideally, we would now prove that the subset: T = { t : Z exp • t 1 λ + t 2 p + t 3 log p λ Γ( λ ) ¶ d ( p,λ ) ‚ I ( p ≥ 0) I ( λ ≥ 0) d ( p,λ ) < ∞} is nonempty and for t ∈ T ( t ): ω ( t ) = log •Z exp t 1 λ + t 2 p + t 3 log p λ Γ( λ ) ¶ I ( p ≥ 0) I ( λ ≥ 0) d ( p,λ ) ¶‚ (b) For the Beta density, we have: p ( x | θ ) ∝ exp • a log( x ) + b log( x ) + log Γ( a + b ) Γ( a )Γ( b ) ¶‚ And hence, a conjugate family is an exponential family on ( a,b ) with natural statistics a,b, Γ( a + b ) Γ( a )Γ( b ) , that is: π ( a,b ) = exp • t 1 a + t 2 b + t 3 log Γ( a + b ) Γ( a )Γ( b ) ¶- ω ( t ) ‚ I ( a ≥ 0) I ( b ≥ 0) 1 which can be normalized to be a density for: T = { t : Z exp • t 1 a + t 2 b + t 3 log Γ( a + b ) Γ( a )Γ( b ) ¶‚ I ( a ≥ 0) I ( b ≥ 0) d ( a,b ) < ∞} by setting: ω ( t ) = log Z exp • t 1 a + t 2 b + t 3 log Γ( a + b ) Γ( a )Γ( b ) ¶‚ I ( a ≥ 0) I ( b ≥ 0) d ( a,b ) ¶ Notice that for t 1 < ,t 2 < 0 and t 3 = 0, this corresponds to the distribution of two independent random variables with exponential distributuons....
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stat210a_2007_hw5_solutions - UC Berkeley Department of...

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