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409Hw02ans - STAT 409 Fall 2009 Homework#2(due Friday...

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STAT 409 Homework #2 Fall 2009 (due Friday, September 11, by 4:00 p.m.) 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( 29 ( 29 ( 29 θ 2 X X ln 1 θ θ ; x x x f x f - = = , x > 1, θ > 1. a) We already know ( Homework 1 ) that the maximum likelihood estimator of θ is = + = n i i x n 1 ln 2 1 θ ˆ . Is θ ˆ a consistent estimator for θ ? Justify your answer . E ( ln X ) = ( 29 - 1 θ 2 1 θ ln ln dx x x x = … = ( 29 1 θ 2 - . By WLLN, = n i i n 1 X ln 1 P E ( ln X ) = ( 29 1 θ 2 - . ( Var ( ln X ) = ( 29 2 1 θ 2 - < ) g ( x ) = x 2 1 + is continuous at ( 29 1 θ 2 - . g ( = n i i n 1 X ln 1 ) = θ ˆ . g ( ( 29 1 θ 2 - ) = θ . θ ˆ P θ . θ ˆ is a consistent estimator for θ .
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b) We already know ( Homework 1 ) that if θ > 2 then the method of moments estimator of θ is 1 1 2 θ ~ - - = x x . Is θ ~ a consistent estimator for θ ? Justify your answer . ( Assume θ > 3 ) . E(X) = ( 29 ( 29 ( 29 ( 29 2 2 1 θ 2 X 2 θ 1 θ ln 1 θ - - = - = - dx x x x dx x f x . By WLLN, X P μ = E ( X ) = ( ( 29 2 2 2 θ 1 θ - - . ( Var ( X ) = σ 2 = ( ( ( 29 ( 29 2 4 2 2 3 θ 2 θ 7 θ 8 θ 2 1 θ - - + - - < ) g ( x ) = 1 1 2 - - x x is continuous at ( ( 29 2 2 2 θ 1 θ - - . g ( X ) = θ ~ . g ( ( ( 29 2 2 2 θ 1 θ - - ) = θ . θ ~ P θ . θ ~ is a consistent estimator for θ .
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2. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( 29 θ 2 θ x e x x f - = x > 0 θ > 0. a) We already know ( Homework 1 ) that the maximum likelihood estimator of θ is θ ˆ = = n i i n 1 X . Is θ ˆ a consistent estimator for θ ? Justify your answer . Hint: Find E ( X ) . ( X E = - 0 θ 2 θ dx x x x e u = x du = 2 x dx = - 0 θ θ du u u e = θ 1 . By WLLN, ( 29 θ 1 X E X 1 1 = = P n i i n . Since g ( x ) = 1 / x is continuous at θ 1 , θ ˆ = = θ 1 X 1 1 g n g P n i i = θ . θ ˆ is a consistent estimator for θ .
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b) We already know ( Homework 1 ) that the method of moments estimator of θ is = = = n i i n 1 X 2 X 2 θ ~ . Is θ ~ a consistent estimator for θ ?
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