410Hw09ans - STAT 410 Homework#9(due Friday October 30 by...

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STAT 410 Homework #9 Fall 2009 (due Friday, October 30, by 3:00 p.m.) Warm-up: 4.2.3 By Chebyshev’s Inequality, P ( | W n μ | ε ) 2 2 W ε σ n = 2 ε p n b 0 as n for all ε > 0. Therefore, μ W P n . 1 – 2. If the random variable Y denotes an individual’s income, Pareto’s law claims that P ( Y y ) = θ y k , where k is the entire population’s minimum income. It follows that f Y ( y ) = 1 θ θ 1 θ y k , y k ; θ 1. The income information has been collected on a random sample of n individuals: Y 1 , Y 2 , … , Y n . 1. Assume k is known. Hint: Recall that k n n n i i ln 1 Y ln θ ˆ - = = = k ln ln Y 1 - and k - = Y Y θ ~ .
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a) Show that the maximum likelihood estimator θ ˆ is a consistent estimator of θ . Hint: Find E ( ln Y ) first. E ( ln Y ) = + k dy y k y 1 θ θ 1 θ ln = - - k dy y y k 1 θ θ ln θ = ( 29 k y y y k - - - - - θ 1 θ 1 θ θ 2 θ θ ln = θ 1 ln + k . By WLLN, ( 29 θ 1 Y E Y ln 1 ln ln 1 + = = k n P n i i . a P n X , g is continuous at a ( 29 ( 29 X a g g P n Consider g ( x ) = k x ln 1 - . Then g ( x ) is continuous at θ 1 ln + = k a . θ ˆ Y ln 1 1 = = n i i n g ( θ = a g . θ θ ˆ P . θ ˆ is a consistent estimator of θ . b) Show that the method of moments estimator θ ~ is a consistent estimator of θ . ( 29 ( 29 1 θ θ θ 1 θ θ Y E θ θ 1 θ θ Y ; - = = = = - + - k dy y k dy y k y dy y f y k k . By WLLN, ( 29 1 θ θ Y E Y - = k P . a P n X , g is continuous at a ( 29 ( 29 X a g g P n
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Consider g ( x ) = k x x - . Then g ( x ) is continuous at 1 θ θ - = k a .
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