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hw3_stat210a

hw3_stat210a - UC Berkeley Department of Statistics STAT...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 3 Fall 2006 Issued: Thursday, September 14, 2006 Due: Thursday, September 21, 2006 Graded Problem 3.1 The inverse Gaussian distribution IG( λ, μ ) has density function λ 2 π exp( λμ ) x - 3 / 2 exp - 1 2 ( λ x + μx ) , x > 0 , λ > 0 , μ > 0 . (a) Show that this density constitutes an exponential family. (b) Show that this family is a scale family. (c) Show that the statistics ¯ X = 1 n n i =1 X i and S = n i =1 ( 1 /X i - 1 / ¯ X ) are complete and sufficient statistics. Problem 3.2 Determine the natural parameter space of the associated exponential family of dimension one with X = R , T ( x ) = x , and (a) h ( x ) = exp( - x 2 ). (b) h ( x ) = exp( -| x | ). (c) h ( x ) = exp( -| x | ) / (1 + x 2 ). Problem 3.3 Let Z be distributed according to a standard normal. For a given θ R , define G θ ( x ) = P ( Z x | Z > θ ) (a) Prove that G is a cdf for any θ . (b) Prove that the distribution G belongs to the exponential family. (c) Find the moment generating function of the distribution G and compute the mean and variance of G .

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