03_31 - θ , since by WLLN & 1 1 X ² + = → P , and g...

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STAT 410 Examples for 03/31/2008 Spring 2008 Central Limit Theorem X 1 , X 2 , … , X n are i.i.d. with mean μ and variance σ 2 . ( ) X - n n = X 1 n n n i i ± ² ³ ´ µ - = Z D , Z ~ N ( 0, 1 ). Theorem 4.3.9 ( ) X - n n ( ) 2 , 0 N D g ( x ) is differentiable t θ and g ' ( θ ) 0 · ( ) ( ) ( ) X g g n n - ( ) ( ) ± ² ³ ´ µ 2 2 , 0 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 f ( x ; θ ) = ¸ ¸ ¹ ¸ ¸ º » - otherwise 0 1 0 1 1 x x 0 < θ < . a) Recall that the maximum likelihood estimator of θ , = - = n i i n 1 X 1 ln ˆ , is a consistent estimator of θ , since by WLLN ( ) ln ln 1 X E X 1 - = = P n i i n . Show that ˆ is asymptotically normally distributed ( as n ) . Find the parameters.

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Let Y i = – ln X i , i = 1, 2, … , n . Then E ( Y 1 ) = θ . E ( Y 1 2 ) = ( ) ± ± ± ² ³ ´ ´ ´ µ - - 1 0 1 2 1 ln dx x x = 2 θ 2 . Var ( Y 1 ) = θ 2 . By CLT, ( ) ( ) 2 Y Y ± ² , 0 Y N n D - . · ( ) ( ) 2 , 0 ˆ N n D - . · ± ± ² ³ ´ ´ µ 2 N ~ ˆ n , ( as n ). b) Recall that the method of moments estimator of θ , X X 1 ~ - = , is a consistent estimator of
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Unformatted text preview: θ , since by WLLN & 1 1 X ² + = → P , and g ( x ) = x x-1 is continuous at & 1 1 + . Show that & ~ is asymptotically normally distributed ( as n → ∞ ) . Find the parameters. E ( X 1 2 ) = & ± ± ± ² ³ ´ ´ ´ µ ¶-⋅ ⋅ 1 & & 1 2 & 1 dx x x = & 2 1 1 + . σ 2 = Var ( X 1 ) = ( )( ) 2 2 & & & 1 2 1 + + . By CLT, ( ) ( ) 2 ± ² , X N n D →-. Since g ( x ) = x x-1 is differentiable at & 1 1 ² + = , g ' ( μ ) = – ( 1 + θ ) 2 ≠ 0, by Theorem 4.3.9, ( ) ( ) ( ) ( ) ( ) ( )( ) & & ± ² ³ ³ ´ µ + + +-→-⋅ 2 2 2 2 & & & & N 1 2 1 1 X ± , D g g n . ¶ ( ) ( ) ( ) & & ± ² ³ ³ ´ µ + + →-& & & N & & ~ 2 2 2 1 1 , D n . ¶ ( ) ( ) & & ± ² ³ ³ ´ µ + + & & & & N & ~ 2 2 2 1 1 ~ n , ( as n → ∞ )....
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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03_31 - θ , since by WLLN & 1 1 X ² + = → P , and g...

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