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Unformatted text preview: Statistics 610: Homework 5 Moulinath Banerjee University of Michigan Announcement: The Homework carries 120 points. Max possible score is 120. Due December 1. (1) FInd a test function φ that maximizes E φ ( X ) subject to E ( X 2 ) = E φ (1- X 2 ) = 1 / 2 invoking the generalized Neyman-Pearson lemma. Show your calculations as explicitly as possible. (2) Consider the following testing problem: X 1 ,X 2 ,...,X n i.i.d.Exp( λ ), with rate parameter λ . (i) Consider testing λ = λ against λ = λ 1 for some λ 1 < λ . Let f 1 ( X 1 ,X 2 ,...,X n ) denote the joint density of the data under λ 1 and f ( X 1 ,X 2 ,...,X n ) that under H . Show that f 1 /f is increasing in X n . (ii) Consider the test that rejects when X n is larger than q 2 n (1- α ) / 2 λ n , where q k ( β ) is the β ’th quantile of the χ 2 k distribution. Show that this the UMP level α test for testing H : λ = λ against λ < λ ....
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This note was uploaded on 04/14/2010 for the course STATS 610 taught by Professor Moulib during the Fall '09 term at University of Michigan.
- Fall '09