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hw6_stat210a

# hw6_stat210a - UC Berkeley Department of Statistics STAT...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 6 Fall 2006 Issued: Thursday, October 5, 2006 Due: Thursday, October 12, 2006 Problem 6.1 In the inverse binomial sampling procedure, N is a random variable representing the number of trials required to observe x successes in a total of N + x Bernoulli trials (with parameter θ ). (a) Show that the best (minimum variance) unbiased estimator of θ is given by δ * ( N ) = ( x - 1) / ( N + x - 1). (b) Show that the information contained in N about θ is I ( θ ) = x/ [ θ 2 (1 - θ )]. (c) Show that var( δ * ) > 1 /I ( θ ). Problem 6.2 Necessity of regularity conditions: Suppose that X Uni[0 , θ ], and note that ψ ( x ; θ ) = ∂θ log p ( x ; θ ) exists for all θ > x . We can thus define moments of the random variable ψ ( X ; θ ). Show that (a) E [ ψ ( X, θ )] = - 1 θ = 0. (b) var( ψ ( X ; θ )) = 0 and I ( θ ) = E [ ψ 2 ( X ; θ )] = 1 2 . (c) The estimator δ ( X ) = 2 X is unbiased for θ , and moreover var( δ ) = θ 2 3 < 1 /I ( θ ). Problem 6.3 Consider a scale family 1 θ f ( x/θ ) , θ > 0 where f is some fixed density function.

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