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# 11_09ans - STAT 410 Examples for Fall 2009 Central Limit...

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STAT 410 Examples for 11/09/2009 Fall 2009 Central Limit Theorem X 1 , X 2 , … , X n are i.i.d. with mean μ and variance σ 2 . ( 29 X - n n = X 1 n n n i i - = Z D , Z ~ N ( 0, 1 ). Theorem 4.3.9 ( 29 θ X - n n ( 29 2 , 0 N D g ( x ) is differentiable t θ and g ' ( θ ) 0 ( 29 ( 29 ( 29 X θ g g n n - ( 29 ( 29 2 2 , 0 θ N ' g D 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( 29 ( 29 ( 29 θ 2 X X ln 1 θ θ ; x x x f x f - = = , x > 1, θ > 1. a) Recall ( Homework 8 ) that the maximum likelihood estimator of θ is X 2 1 X 2 1 θ ˆ ln ln 1 + = + = = n i i n . Recall ( Homework 9 ) that θ ˆ is a consistent estimator of θ . Show that θ ˆ is asymptotically normally distributed ( as n ). Find the parameters. E [ ( ln X ) 2 ] = ( 29 ( 29 - 1 θ 2 2 1 θ ln ln dx x x x = ( 29 2 1 θ 6 - . Var ( ln X ) = ( 29 ( 29 2 2 1 θ 2 1 θ 6 - - - = ( 29 2 1 θ 2 - . By CLT, ( 29 - - = 1 θ 2 X ln 1 1 n i i n n D N ( 0, ( 29 2 1 θ 2 - ).

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g ( x ) = x 2 1 + . g ' ( x ) = 2 2 x - . g
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11_09ans - STAT 410 Examples for Fall 2009 Central Limit...

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