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HW02 - UC Berkeley Department of Statistics STAT 210A...

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UC Berkeley, Department of Statistics Fall, 2009 STAT 210A: Theoretical Statistics HW#2 Due: Tuesday, September 15, 2009 Sufficient Statistic 2.1. Suppose that , 1,..., i X i n = are i.i.d. Poisson random variables with parameter θ . Show that 1 n i i T X = = is sufficient by computing the conditional distribution given T t = . 2.2. Let 1 ,..., n X X be independent Bernoulli variables with ( 1) i i p P X = = for 1,..., i n = . Let 1 ,..., n t t be a sequence of known constants related to i p by: log( ) 1 i i i p t p α β = + - where α and β are unknown parameters. Determine a sufficient statistic for the family of joint distributions indexed by ( , ) θ α β = . 2.3. Suppose that 1 ,..., n X X are a collection of (non-i.i.d.) Bernoulli random variables with the following joint distribution: 1 1 ( 1) P X θ = = 11 1 1 1 10 1 if 1 ( 1| ,..., ) if 0 i i i i X P X X X X θ θ - - - = = = = where 1 11 10 ( , , ) θ θ θ θ = . Find a three-dimensional sufficient statistic for θ . Minimal Sufficiency 2.4. Consider the family ( , ) N θ θ for θ > 0. (a) Show that the statistic 2 1 1 ( , ) n n i i i i U X X = = = is sufficient. (b) Is U minimal sufficient? Why or why not? 2.5. Let X and Y be independent variables, with X drawn from a Poisson distribution with parameter θ and Y drawn from a Poisson distribution with mean
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