# sol1 - PSTAT 120C: Assignment #1 Solutions April 17, 2006...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## This note was uploaded on 04/09/2008 for the course PSTAT 120c taught by Professor Idk during the Spring '07 term at UCSB.

### Page1 / 5

sol1 - PSTAT 120C: Assignment #1 Solutions April 17, 2006...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online