Hw08ans - STAT 410 (due Friday, March 14, by 3:00 p.m.)...

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STAT 410 Spring 2008 Homework #8 (due Friday, March 14, by 3:00 p.m.) 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( ) ( ) ( ) 2 X X ln 1 ; x x x f x f - = = , x > 1, θ > 1. a) Find the maximum likelihood estimator ˆ of θ . L( θ ) = ( ) = - n i i i x x 1 2 ln 1 . ln L( θ ) = ( ) = = - + - n i i n i i x x n 1 1 ln ln ln 1 2 ln . ( ) = - - = n i i d d x n 1 ln 1 2 L = 0. ± = + = n i i x n 1 ln 2 1 ˆ . b) Suppose θ > 2. Find the method of moments estimator ~ of θ . E(X) = ( ) ( ) ( ) ( ) 2 2 1 2 X 2 1 ln 1 & - - = - = ² ² - dx x x x dx x f x . ( ) ( ) 2 2 1 2 1 1 - - = = = x x n n i i . ± 1 1 2 ~ - - = x x .
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If the random variable Y denotes an individual’s income, Pareto’s law claims that P ( Y y ) = ± ² ³ ´ µ y k , where k is the entire population’s minimum income. It follows that f Y ( y ) = 1 1 + ± ² ³ ³ ´ µ 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 . Assume k is known. a) Find the maximum likelihood estimator ˆ of θ . Likelihood function: L( θ ) = ( ) ( ) 1 1 1 Y Y Y ; + - = = ± ² ³ ³ ´ µ = n i n n n i i i k f , Y i k , 1 i n . ln L( θ ) = ( ) = + - + n i i k n n 1 Y ln 1 ln ln . ( ) ( ) = - + = n i i d d k n n 1 Y ln ˆ ˆ L ln ln = 0. · k n n n i i ln 1 Y ln ˆ - = = . b) Find the method of moments estimator ~ of θ . ( ) ( ) 1 1 Y E 1 Y ; - = = ± ² ³ ³ ³ ´ µ ± ² ³ ³ ´ µ = = ¸ ¸ ¸ - + - k dx y k dx y k y dx y f y k k . 1
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This homework help was uploaded on 04/13/2008 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Hw08ans - STAT 410 (due Friday, March 14, by 3:00 p.m.)...

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