10_24 - STAT 410 Examples for 10/24/2011 Fall 2011 1. Let X...

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STAT 410 Examples for 10/24/2011 Fall 2011 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) Recall that the method of moments estimator of θ , X X 1 θ ~ - = , is a consistent estimator of θ , since by WLLN θ 1 1 X μ + = P , and g ( x ) = x x - 1 is continuous at θ 1 1 + . Show that θ ~ is asymptotically normally distributed ( as n ) . Find the parameters. E ( X 2 ) = - 1 0 θ θ 1 2 θ 1 dx x x = θ 2 1 1 + . σ 2 = Var ( X ) = ( )( ) 2 2 θ θ θ 1 2 1 + + . By CLT, ( ) ( ) 2 σ μ , 0 X N n D - . Since g ( x ) = x x - 1 is differentiable at θ 1 1 μ + = , g ' ( μ ) = – ( 1 + θ ) 2 0,
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by Theorem 4.3.9, ( ) ( ) ( ) ( ) ( ) ( )( ) + + + - - 2 2 2 2 θ θ θ θ N 1 2 1 1 0 X μ , D g g n . ( ) ( ) ( ) + + - θ θ θ N θ θ ~ 2 2 2 1 1 0 , D n . For large n , ( ) ( ) + + θ θ θ θ N θ ~ 2 2 2 1 1 ~ n , . b) Recall that the maximum likelihood estimator of θ , = - = n i i n 1 X 1 ln θ ˆ , is a consistent estimator of θ , since by WLLN ( ) θ ln ln 1 X E X 1 - = = P n i i n . Show that θ ˆ is asymptotically normally distributed ( as n ) . Find the parameters. Let Y i = – ln X i , i = 1, 2, … , n . Then E ( Y ) = θ . E ( Y 2 ) = ( ) - - 1 0 θ θ 1 2 θ 1 ln dx x x = 2 θ 2 . Var ( Y ) = θ 2 . By CLT, ( ) ( ) 2 Y Y σ μ , 0 Y N n D - . ( ) ( ) 2 θ θ , 0 θ ˆ N n D - . For large n , 2 θ θ N ~ θ ˆ n , .
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c) Construct a 100 ( 1 – α ) % confidence interval for θ . For large n , ( ) ( ) + + θ θ θ θ N θ ~ 2 2 2 1 1 ~ n , . X X 1 θ ~ - = ( ) ( ) n z θ ~ θ ~ θ ~ θ ~ 2 1 1 α 2 + + ± would have an approximate 100 ( 1 – α ) % confidence level for large n . OR For large n , 2 θ θ N ~ θ ˆ n , . = -
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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_24 - STAT 410 Examples for 10/24/2011 Fall 2011 1. Let X...

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