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 β . b) (7) Obtain the maximum likelihood estimator of β , β ˆ . L ( β ) = ∏ =- n i x i i e x 1 4 β 12 β = ∑ =-∏ = n i i x n i i n n e x 1 4 1 β 12 β ln L ( β ) = ∑ ∑ = = ⋅ ⋅ ⋅-+-n i i n i i x x n n 1 1 12 4 β ln ln β ln ( ln L ( β ) ) ' = ∑ =-n i i x n 1 4 β = 0 ⇒ β ˆ = ∑ = 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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