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 ; ) = exp[ x log - log (- log(1- ))] I ( x N ) x so p belongs to an exponential family with h ( x ) = I ( x N ) x , T ( x ) = x , ( ) = log and B ( ) = log(- log(1- )). Hence, = exp( ) and A ( ) = B (exp( )) = log (- log(1- exp( )). It follows that: E ( T ( X )) = E X = d A ( ) d =- 1 log(1- exp( )) exp( ) 1- exp( ) =- 1 log(1- ) (1- ) var ( T ( X )) = var X = d 2 A ( ) d 2 =- exp( ) log(1- exp( ))[1- exp( )] 2 1 + exp( ) log(1- exp( )) =- log(1- )[1- ] 2 1 + log(1- ) Problem 5.2 a) For the Gamma density, we have: p ( x | ) exp log( x )- px + log p ( ) I ( p 0) I ( 0) And hence, a conjugate family is an exponential family on ( ,p ) with natural statistics ,p, log p ( ) , that is: ( ,p ) = exp t 1 + t 2 p + t 3 log p ( )- ( t ) I ( p 0) I ( 0) Ideally, we would now prove that the subset: T = { t : Z exp t 1 + t 2 p + t 3 log p ( ) d ( p, ) I ( p 0) I ( 0) d ( p, ) < } is nonempty and for t T...
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stat210a_HW05 - UC Berkeley Department of Statistics STAT...

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