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Stat610-Homework5 - Statistics 610 Homework 5 Moulinath...

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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 0 against λ = λ 1 for some λ 1 < λ 0 . Let f 1 ( X 1 , X 2 , . . . , X n ) denote the joint density of the data under λ 1 and f 0 ( X 1 , X 2 , . . . , X n ) that under H 0 . Show that f 1 /f 0 is increasing in X n . (ii) Consider the test that rejects when X n is larger than q 2 n (1 - α ) / 2 λ 0 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 0 : λ = λ 0 against λ < λ 0 . (iii) Show that for any λ , β ( λ ) = P λ ( X n c α,n ) = 1 - F 2 n λ λ 0 ˙ q 2 n (1 - α ) ! , where F 2 n is the distribution function of χ 2 2 n and conclude that this is de- creasing in λ .
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