409Quiz1Aans - - . K ( x ) = x . Y = ( ) = n i i 1 X K = =...

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STAT 409 Fall 2011 Version A 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 ( ) x e x x f β 12 4 X β β ; - = , x > 0, β > 0. a) (3) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for β . f ( x 1 , x 2 , x n ; β ) = f X ( x 1 ; β ) f X ( x 2 ; β ) f X ( x n ; β ) = = = - n i i x n n x n i i e 1 4 12 1 β β . By Factorization Theorem, Y = = n i i 1 X is a sufficient statistic for β . OR f X ( x ; β ) = { } n ln β ln β exp l 12 4 x x + - +
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Unformatted text preview: - . K ( x ) = x . Y = ( ) = n i i 1 X K = = n i i 1 X is a sufficient statistic for . b) (7) Obtain the maximum likelihood estimator of , . L ( ) = =- n i x i i e x 1 4 12 = =- = n i i x n i i n n e x 1 4 1 12 ln L ( ) = = = -+-n i i n i i x x n n 1 1 12 4 ln ln ln ( ln L ( ) ) ' = =-n i i x n 1 4 = 0 = = n i i n 1 X 4 ....
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409Quiz1Aans - - . K ( x ) = x . Y = ( ) = n i i 1 X K = =...

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