409Hw06 - STAT 409 Fall 2009 Homework#6(due Friday October...

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STAT 409 Homework #6 Fall 2009 (due Friday, October 9, by 4:00 p.m.) 1. Let a > 0, θ > 0 and let X 1 , X 2 , … , X n be a random sample of size n from a distribution with probability density function ( 29 ( 29 1 X X θ θ ; - = = a a x a x f x f , 0 < x < θ . Suppose a is known. a) Find the sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for θ . b) Find the method of moments estimator θ ~ of θ . 2. Let a > 0, θ > 0 and let X 1 , X 2 , … , X n be a random sample of size n from a distribution with probability density function ( 29 ( 29 1 X X θ θ ; - = = a a x a x f x f , 0 < x < θ . Suppose a is known. a) Find the maximum likelihood estimator θ ˆ of θ . b) Is θ ˆ a consistent estimator of θ ? Justify your answer . Hint: Let ε > 0. Find P ( | θ ˆ θ | ε ) = P ( θ ˆ θ ε ) + P ( θ ˆ θ + ε ). 3. Let a > 0, θ > 0 and let X 1 , X 2 , … , X n be a random sample of size n from a distribution with probability density function ( 29 ( 29

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409Hw06 - STAT 409 Fall 2009 Homework#6(due Friday October...

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