stat210a_HW05

# stat210a_HW05 - UC Berkeley Department of Statistics STAT...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5 - Solutions Fall 2008 Issued: Tuesday, September 29, 2009 Due: Tuesday, October 6, 2009 Problem 5.1 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 )) =- θ log(1- θ )[1- θ ] 2 1 + θ log(1- θ ) Problem 5.2 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...
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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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stat210a_HW05 - UC Berkeley Department of Statistics STAT...

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