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HW05 - 2 2(Hints read pages 69-70 of the Keener book...

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UC Berkeley, Department of Statistics Fall, 2009 STAT 210A: Introduction to Mathematical Statistics HW#5 Due: Tuesday, October 06, 2009 4.1. By Taylor series expansion, we have the identity 1 log(1 ) x x x θ θ = - - = , which is valid for all (0,1) θ . From this fact, the quantity ( , ) , 1,2,... log(1 ) x p x x x θ θ θ = = - - defines a valid probability mass function for all (0,1) θ . Compute the mean and variance of a random variable X with this log series distribution. 4.2. Find conjugate distributions for the following family of distributions: a) the gamma family of distribution functions ( , ) p λ Γ b) the beta family of distribution functions ( , ) p β λ c) the family defined on the parameters 1 β and 2 β by the linear regression: 1 2 i i y x β β ε = + + with 2 ~ (0, ) N ε σ and 2 σ known. d) the multinomial distribution 4.3. Let 1 ,..., n X X be iid from 2 ( , ) N μ σ , μ and 0 σ > both unkown. a) find the Uniformly Minimum Variance Unbiased (UMVU) estimator of 2 μ . b) find the UMVU estimator of
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Unformatted text preview: 2 2 / (Hints: read pages 69-70 of the Keener book; available at bspace) 4.4. Let 1 Z and 2 Z be independent standard normal random variables. Find 1 2 | / | E Z Z . (Hints: again use the materials on the pages 69-70 of the Keener book) 4.5. Regularity Conditions are Needed for the information inequality . Let (0, ) X U ∼ be the uniform distribution on (0, ) . Note that log ( , ) p x is differentiable for all x > , that is, with probability 1 for each , and we can thus define moments of / log ( , ) T p X = ∂ ∂ . Show that, however, a) 1 log ( , ) E p X ∂ = -≠ ∂ . b) log ( , ) Var p X ∂ = ∂ and 2 2 ( ) 1/ I ET = = . c) 2 X is unbiased for and has variance 2 2 (1/ 3) (1/ ( )) I < = ....
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