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# 04_02_2 - STAT 410 Examples for Spring 2008 Central Limit...

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STAT 410 Examples for 04/02/2008 Spring 2008 Central Limit Theorem X 1 , X 2 , … , X n are i.i.d. with mean μ and variance σ 2 . ( ) σ μ X - n n = σ μ 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 σ D g ( x ) is differentiable t θ and g ' ( θ ) 0 · ( ) ( ) ( ) X ° g g n n - ( ) ( ) ° ± ² ³ ´ µ 2 2 , 0 ° N σ ' g D 1. Let X 1 , X 2 , … , X n be a random sample of size n from a Geometric ( p ) distribution ( the number of independent trials until the first “success” ) . That is, P ( X 1 = k ) = ( 1 – p ) k – 1 p , k = 1, 2, 3, … . Recall that the maximum likelihood estimator of p , p ˆ , and the method of moments estimator of p , p ~ , are equal and p ˆ = p ~ = X 1 X 1 = = n i i n . a) Use WLLN ( Theorem 4.2.1 ) and Theorem 4.2.4 to show that p ˆ is a consistent estimator for p ( as n ) . By WLLN, ( ) p P 1 X E X ± = = . Since g ( x ) = 1 / x is continuous at 1 / p , p ˆ = ( ) ° ± ² ³ ´ µ 1 X p g g P = p . · p ˆ is a consistent estimator for p .

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b) Use CLT ( Theorem 4.4.1 ) and Theorem 4.3.9 to show that p ˆ is asymptotically normally distributed ( as n ) . Find the parameters. Var ( X ) = 2 1 p p - . By CLT, ° ° ± ² ³ ³ ´ µ - ° ± ² ³ ´ µ - 2 1 0 1 X N p p p n , D . Since g ( x ) = 1 / x is differentiable at 1 / p , g ' ( 1 / p ) = – p 2
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04_02_2 - STAT 410 Examples for Spring 2008 Central Limit...

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