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Unformatted text preview: STAT 410 Homework #9 Fall 2009 (due Friday, October 30, by 3:00 p.m.) Warmup: 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 ~ . 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 ( 29 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 → Consider g ( x ) = k x x . Then g ( x ) is continuous at 1 θ θ = k a ....
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This note was uploaded on 04/15/2010 for the course STAT 410 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Monrad

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