10_07 - STAT 410 Examples for 10/07/2011 Fall 2011 Def An...

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Unformatted text preview: STAT 410 Examples for 10/07/2011 Fall 2011 Def An estimator is said to be unbiased for if E( ) = for all . 3. Let X 1 , X 2 , , X n be a random sample of size n from the distribution with probability density function f ( x ; ) = - otherwise 1 1 1 x x 0 < < . Recall: The method of moments estimator of is ~ = 1 X 1 X X 1- =- , the maximum likelihood estimator of is = = - n i i n 1 X ln 1 . d) Is unbiased for ? That is, does E( ) equal ? ( ) ( ) = =- - 1 1 X 1 1 ln ln X ln E ; dx x x dx x f x . Integration by parts: - = b a b a du v a b v u dv u Choice of u : L ogarithmic A lgebraic T rigonometric E xponential u = ln x , dv = dx x dx x 1 1 1 1 1-- = , du = dx x 1 , v = 1 x . ( ) - = = - 1 1 1 1 1 1 1 1 ln 1 ln X ln E dx x x x x dx x x = 1 1 1 1 1 1 1 1 1 1- = - =- = - - x dx x dx x x . Therefore, ( ) ( ) ( ) = =-- =- = n i n i i n n 1 1 1 X ln E 1 E = , that is, is an unbiased estimator for . OR F X ( x ) = x 1 / , 0 < x < 1....
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This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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10_07 - STAT 410 Examples for 10/07/2011 Fall 2011 Def An...

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