stat210a_2007_hw3_solutions

# stat210a_2007_hw3_solutions - UC Berkeley Department of...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 3- Solutions Fall 2007 Issued: Thursday, September 13 Due: Thursday, September 20 Problem 3.1 (a) Let F X be distribution of X . Then, F X ( x ) = 1- e- λx for all λ > 0. Thus, F Y ( y ) = P ( Y ≤ y ) = P ( X ≤ y λ ) = F X ( y λ ) = 1- e- y . i.e. Y has an exponential distribution with failure rate 1. (b) Because X is 1-dimensional full rank exponential family, ¯ X is a complete sufficient statistics. Let Y i = λX i , i = 1 ,...,n . Then, X 2 1 + ··· + X 2 n ¯ X 2 = Y 2 1 + ··· + Y 2 n ¯ Y 2 : not depend on λ . Thus, X 2 1 + ··· + X 2 n ¯ X 2 is an ancillary statistics. By Basu’s theorem, ¯ X and X 2 1 + ··· + X 2 n ¯ X 2 are independent. (c) X (1) X ( n ) = Y (1) Y ( n ) : not depend on λ . By Basu’s theorem, ¯ X and X (1) X ( n ) are independent. Problem 3.2 To determine the natural parameter space of each of the families, we must determine the set A T,h = { η : R h ( x )exp[ ηT ( x )] dx < ∞} . (a) In this case A h,T = R . To prove that, notice that: Z exp £ ηx- x 2 / dx = exp η 2 4 ¶Z exp •- ‡ η 2 + x · 2 ‚ dx = √ 2exp η 2 4 ¶Z exp •- y 2 2 ‚ dy = 2 √ π exp η 2 4 ¶Z exp 1 √ 2 π •- y 2 2 ‚ dy = 2 √ π exp η 2 4 ¶ < ∞ , ∀ η ∈ R (b) First, notice that: C ( η ) = Z exp[ ηx- | x | ] dx = Z x> exp[( η- 1) x ] dx + Z x< exp[( η + 1) x ] dx 1 If η <- 1, lim x ↓-∞ exp[( η + 1) x ] > 0 so the integral over x < 0 diverges. Likewise, if η > 1, lim x ↑ + ∞ exp[( η + 1) x ] > 0 so the integral over x > 0 diverges. Now, if- 1 < η < 1, both integrals can be rewritten as R ∞ exp[- ax ] dx with a > 0 which is finite for any value of a . Now if η = 1 ( alt. η =- 1), the integral over the positive branch (alt. negative branch) diverges. As a result the natural parameter space is given by η ∈ (- 1 , 1)....
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stat210a_2007_hw3_solutions - UC Berkeley Department of...

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