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Unformatted text preview: PSTAT 120C: Assignment #1 Solutions April 17, 2006 1. Since T is a sufficient statistic for estimating By the factorization theorem we can factor the likelihood as follows. L ( ) = g ( ,T ) h ( X 1 ,..,X n ) where the h function is the same for all . The GLRT test statistic is = max L ( ; X 1 ,...,X n ) max L ( ; X 1 ,...,X n ) = max g ( ,T ) h ( X 1 ,...,X n ) max g ( ,T ) h ( X 1 ,...,X n ) = max g ( ,T ) max g ( ,T ) because h is the same for all and can be pulled outside of the maximization. Hence the Likelihood Ratio Test statistics for testing H : versus H a : / is a function of the g s which are functions of T . 2. (a) For the null hypothesis, the probability p = 1 / 2 so max L ( p ) = n x 2- n . For the whole parameter space = [0 , 1], we have shown before that p = X/n is the maximum likelihood estimator. max L ( p ) = n x x n x 1- x n n- x . The ratio is therefore = ( n x ) 2- n ( n x )( x n ) x ( 1- x n ) n- x = 2- n x n- x n- x n- ( n- x ) . (b) If this < k then * ( x ) = x n x n n- x n n- x n > 1 2 k- 1 /n ....
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This note was uploaded on 04/09/2008 for the course PSTAT 120c taught by Professor Idk during the Spring '07 term at UCSB.
- Spring '07