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Unformatted text preview: UC Berkeley, Department of Statistics Fall, 2009 STAT 210A: Introduction to Mathematical Statistics HW#4 Due: Tuesday, September 29, 2009 4.1. Let 1 ,.., n X X be independent with ~ ( ,1) i i X N t , where 1 ,..., n t t are a sequence of known constants (not all zero). (a) Show that the least squares estimator, c 2 / i i i t X t = is complete sufficient for the family of joint distributions. (b) Use Basus theorem to show that c and c 2 ( ) i i X t - are independent. 4.2. The inverse Gaussian distribution ( , ) IG has density function 3/ 2 1 exp( ) exp ( ) , 0, 0, 0. 2 2 x x x x - - + > > > (a) Show that this density constitutes an exponential family. (b) Show that the statistic T = ( 1 1 n i i X X n = = , 1 1 1 ( ) n i i S X X = =- ) is complete and sufficient. 4.3. Determine the natural parameter space of the associated exponential family of dimension one with = R , ( ) T x x = and (a) h ( x ) = exp(-...
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This note was uploaded on 10/17/2009 for the course STAT 210a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.
- Fall '08