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HW2 - Let μ = E X(a(10 pts Find the UMP level-α test for...

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STA131C, Spring 2007 Prof. W. Polonik HW # 2 due: April 18, 2007 in class 1. (15 pts) Let X 1 , . . . , X n iid N ( μ, 1). Show that in this model hatwide θ = ( X n ) 2 - 1 n is the UMVUE for θ = μ 2 . 2. Let X 1 , . . . , X n iid Uniform(0 , θ ) , θ > 0 . (a) (10 pts) Find the Cramer-Rao lower bound for unbiased estimators of θ. (b) (10 pts) Let X ( n ) = max( X 1 , . . . , X n ) . Show that hatwide θ = n +1 n X ( n ) is an unbiased estimator of θ that has uniformly smaller variance than the Cram´ er-Rao lower bound (i.e. the variance of hatwide θ is smaller than the Cram´ er-Rao lower bound for ever value of θ > 0 and for any n 1). (c) (5 pts) Do you have an explanation for what happened in (b)? 3. Let X 1 , . . . , X n be iid. from an exponential distribution with pdf f ( x, θ ) = θe - θx
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Unformatted text preview: . Let μ = E( X ) . (a) (10 pts) Find the UMP level-α test for testing H : μ ≤ μ versus H 1 : μ > μ , some a given μ > . HINT: Use that if X i have pdf f ( x, θ ) then 2 θ ∑ n i =1 X i ∼ χ 2 2 n . (10 pts extra credit for a proof of this fact.) (b) (10 pts) Find an expression for the power of the UMP-test from part (a) at a given μ 1 with μ 1 > μ . (c) (10 pts) Use the central limit theorem to ±nd approximate values for the critical value of the UMP-test. (d) (10 pts) Use parts (b) and (c) to in order to ±nd the power for n = 22 , μ = 10 , μ 1 = 12 . 5 and α = 0 . 05 ....
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