Dr. Hackney STA Solutions pg 38

Dr. Hackney STA Solutions pg 38 - Second Edition 3-11(iii...

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Second Edition 3-11 (iii) α, β unknown, h ( x ) = I [0 , 1] ( x ) , c ( α, β ) = 1 B ( α,β ) , w 1 ( α ) = α - 1 , t 1 ( x ) = log x, w 2 ( β ) = β - 1 , t 2 ( x ) = log(1 - x ). d. h ( x ) = 1 x ! I { 0 , 1 , 2 ,... } ( x ) , c ( θ ) = e - θ , w 1 ( θ ) = log θ, t 1 ( x ) = x . e. h ( x ) = x - 1 r - 1 I { r,r +1 ,... } ( x ) , c ( p ) = p 1 - p r , w 1 ( p ) = log(1 - p ) , t 1 ( x ) = x . 3.29 a. For the n( μ, σ 2 ) f ( x ) = 1 2 π e - μ 2 / 2 σ 2 σ e - x 2 / 2 σ 2 + xμ/σ 2 , so the natural parameter is ( η 1 , η 2 ) = ( - 1 / 2 σ 2 , μ/σ 2 ) with natural parameter space { ( η 1 2 ): η 1 < 0 , -∞ < η 2 < ∞} . b. For the gamma( α, β ), f ( x ) = 1 Γ( α ) β α e ( α - 1) log x - x/β , so the natural parameter is ( η 1 , η 2 ) = ( α - 1 , - 1 ) with natural parameter space { ( η 1 2 ): η 1 > - 1 2 < 0 } . c. For the beta( α, β ), f ( x ) = Γ( α + β ) Γ( α )Γ( β ) e ( α - 1) log x +( β - 1) log(1 - x ) , so the natural parameter is ( η 1 , η 2 ) = ( α - 1 , β - 1) and the natural parameter space is { ( η 1 2 ): η 1 > - 1 2 > - 1 } . d. For the Poisson f ( x ) = 1 x ! ( e - θ ) e x log θ so the natural parameter is η = log θ and the natural parameter space is { η : -∞ < η < ∞} . e. For the negative binomial(
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