hw4_stat210a_solutions

hw4_stat210a_solutions - UC Berkeley Department of...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 4 Fall 2006 Issued: Thursday, September 21, 2006 Due: Thursday, September 28, 2006 Graded exercises Problem 4.1 First, notice that we can rewrite: p ( x ; θ ) = exp [ x log θ - log ( - log(1 - θ ))] I ( x N ) x so p belongs to an exponential family with h ( x ) = I ( x N ) x , T ( x ) = x , η ( θ ) = log θ and B ( θ ) = log( - log(1 - θ )). Hence, θ = exp( η ) and A ( η ) = B (exp( η )) = log ( - log(1 - exp( η )). It follows that: E θ ( T ( X )) = E θ X = d A ( η ) d η = - 1 log(1 - exp( η )) · exp( η ) 1 - exp( η ) = - 1 log(1 - θ ) · θ (1 - θ ) var θ ( T ( X )) = var θ X = d 2 A ( η ) d η 2 = exp( η ) log(1 - exp( η )) [1 - exp( η )] 2 ± - 1 + exp( η ) log(1 - exp( η )) ² = - θ log(1 - θ ) [1 - θ ] 2 ± 1 - θ log(1 - θ ) ² Problem 4.2 (a) For the Gamma density, we have: p ( x | θ ) exp ³ λ log( x ) - px + log ± p λ Γ( λ ) ² I ( p 0) I ( λ 0) ´ And hence, a conjugate family is an exponential family on ( λ,p ) with natural statistics λ,p, log µ p λ Γ( λ ) , that is: π ( λ,p ) = exp ³ t 1 λ + t 2 p + t 3 log ± p λ Γ( λ ) ² - ω ( t ) ´ I ( p 0) I ( λ 0) Ideally, we would now prove that the subset: T = { t : Z exp ³ t 1 λ + t 2 p + t 3 log ± p λ Γ( λ ) ² d ( p,λ ) ´ I ( p 0) I ( λ 0) d ( p,λ ) < ∞} 1
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is nonempty and for t ∈ T ( t ): ω ( t ) = log ±Z exp ² t 1 λ + t 2 p + t 3 log ² p λ Γ( λ ) ³ I ( p 0) I ( λ 0) d ( p,λ ) ³´ (b) For the Beta density, we have: p ( x | θ ) exp ± a log( x ) + b log( x ) + 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 Γ( a + b ) Γ( a )Γ( b ) - ω ( t ) ´ I ( a 0) I ( b 0) which can be normalized to be a density for: T = { t : Z exp ± t 1 a + t 2 b + t 3 Γ( a + b ) Γ( a )Γ( b ) ´ I ( a 0) I ( b 0) d ( a,b ) < ∞} by setting: ω ( t ) = log ²Z exp ± t 1 a + t 2 b + t 3 Γ( a + b ) Γ( a )Γ( b ) ³ I ( a 0) I ( b 0) d ( a,b ) ´ Notice that for t 1 < 0 ,t 2 < 0 and t 3 = 0, this corresponds to the distribution of two
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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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hw4_stat210a_solutions - UC Berkeley Department of...

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