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409Hw02 - 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 . 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 ) . 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 . 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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