# Hw08 - STAT 410(due Friday March 14 by 3:00 p.m Homework#8...

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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 . b) Suppose θ > 2. Find the method of moments estimator ~ of θ . 2. 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 . b) Find the method of moments estimator ~ of θ . 3. Let Y 1 , Y 2 , … , Y n be a random sample of size n from the Pareto distribution: f Y ( y ) = 1 1 + ± ² ³ ³ ´ µ y k , y

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Hw08 - STAT 410(due Friday March 14 by 3:00 p.m Homework#8...

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