07_13ans - STAT 410 Examples for 07/13/2009 Summer 2009 1....

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STAT 410 Examples for 07/13/2009 Summer 2009 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; θ ) = - otherwise 0 1 0 θ 1 θ θ 1 x x 0 < θ < . a) Obtain the method of moments estimator of θ , θ ~ . ( 29 ( 29 = = - - 1 0 θ θ 1 X θ 1 θ X E ; dx x x dx x f x . = θ 1 1 0 1 1 θ 1 1 θ 1 θ 1 1 θ 1 1 0 θ 1 + = + = + x dx x . θ 1 1 X + = . X X 1 θ ~ - = . b) Obtain the maximum likelihood estimator of θ , θ ˆ . Likelihood function: L( θ ) = ( 29 θ θ 1 1 1 X X θ 1 θ X ; - = = = n i n n i i i f . ln L( θ ) = = = - + - = - + - n i i n i i n n 1 1 X ln 1 θ 1 θ X ln θ θ 1 θ ln ln . ( 29 ( 29 = - - = n i i d d n 1 2 X ln θ ˆ 1 θ ˆ θ ˆ L θ ln = 0. = - = n i i n 1 X ln 1 θ ˆ .
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c) Suppose n = 3, and x 1 = 0.2, x 2 = 0.3, x 3 = 0.5. Compute the values of the method of moments estimate and the maximum likelihood estimate for θ . 3
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This note was uploaded on 10/29/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Summer '08 term at University of Illinois at Urbana–Champaign.

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07_13ans - STAT 410 Examples for 07/13/2009 Summer 2009 1....

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