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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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2. If the random variable Y denotes an individual’s income, Pareto’s law claims that P ( Y y ) = ° ³ ´ µ · ¸ y k , where k is the entire population’s minimum income. It follows that f Y ( y ) = 1 ° ° 1 ° + 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 ; + - = = 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 ; - = = ³ ³ ³ ´ µ · ¸ ³ ³ ´ µ · ¸ = = ² ² ² - + - k dx y k dx y k y dx y f y k k .
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