{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 - ).
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
g ( x ) = x 2 1 + . g ' ( x ) = 2 2 x - . g
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online