409Quiz1Eans - x ) = x 2 . ⇒ Y = ( ) ∑ = n i i 1 X K =...

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STAT 409 Fall 2011 Version E Name ANSWERS . Quiz 1 (10 points) Be sure to show all your work, your partial credit might depend on it. No credit will be given without supporting work. 1. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( ) 2 δ 3 7 4 X δ δ ; x e x x f - = , x > 0, δ > 0. a) (3) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for δ . f ( x 1 , x 2 , x n ; δ ) = f X ( x 1 ; δ ) f X ( x 2 ; δ ) f X ( x n ; δ ) = = = - n i i x n n x n i i e 1 7 4 3 1 2 δ δ . By Factorization Theorem, Y = = n i i 1 2 X is a sufficient statistic for δ . OR f X ( x ; δ ) = { } 2 n ln ln exp l 7 3 4 δ δ x x + - + - . K (
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Unformatted text preview: x ) = x 2 . ⇒ Y = ( ) ∑ = n i i 1 X K = ∑ = n i i 1 2 X is a sufficient statistic for δ . b) (7) Obtain the maximum likelihood estimator of δ , δ ˆ . L ( δ ) = ∏ =- n i x i i e x 1 7 4 2 δ 3 δ = ∑ =-∏ = n i i x n i i n n e x 1 7 4 1 2 δ 3 δ ln L ( δ ) = ∑ ∑ = = ⋅ ⋅ ⋅-+-n i i n i i x x n n 1 2 1 7 3 4 δ δ ln ln ln ( ln L ( δ ) ) ' = ∑ =-n i i x n 1 2 4 δ = 0 ⇒ δ ˆ = ∑ = n i i n 1 2 X 4 ....
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This note was uploaded on 09/14/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.

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409Quiz1Eans - x ) = x 2 . ⇒ Y = ( ) ∑ = n i i 1 X K =...

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