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# HW 8 - STAT 400 Fall 2011 Homework#8(due Friday October 28...

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STAT 400 Fall 2011 Homework #8 (due Friday, October 28, by 3:00 p.m.) From the textbook: 6.1-5 ( ) (a) + method of moments estimator f ( x ; θ ) = ( 1 / θ 2 ) x e x / θ , 0 < x < , 0 < θ < . L ( θ ) = - = = n i i n i i n x x 1 1 2 θ 1 θ 1 exp . ( ) = + - = n i i d d x n 1 2 θ 1 θ 2 θ L θ ln . 2 2 1 θ ˆ 1 x x n n i i = = = . E ( X ) = 2 θ ( Gamma, α = 2 ) = = = n i i x n x 1 2 1 2 θ ~ . 6.1-5 ( ) (b) + method of moments estimator f ( x ; θ ) = ( 1 / 2 θ 3 ) x 2 e x / θ , 0 < x < , 0 < θ < . L ( θ ) = - = = n i i n i i n n x x 1 2 1 3 θ 1 θ 2 1 exp . ( ) = + - = n i i d d x n 1 2 θ 1 θ 3 θ L θ ln . 3 3 1 θ ˆ 1 x x n n i i = = = . E ( X ) = 3 θ ( Gamma, α = 3 ) = = = n i i x n x 1 3 1 3 θ ~ .

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6.1-5 ( ) (c) + method of moments estimator f ( x ; θ ) = ( 1 / 2 ) e | x θ | , < x < , < θ < . L ( θ ) = - - = 1 θ 2 1 exp n i i n x . Need to minimize = - n i i x 1 θ . If n is odd, 2 1 Y θ ˆ + = n ( the middle value in the data set ) . If n is even, ] [ 2 2 2 Y , Y θ ˆ + n n ( any value between the middle two ) . f ( x ; θ ) is symmetric about θ . E ( X ) = θ ( balancing point ) = = = n i i x n x 1 1 θ ~ . 6.1-7 ( ) (a), (b) + method of moments estimator + (c)(i) ( ) < < = - otherwise 0 1 0 1 θ θ x x x f 0 < θ < . a)
b) Likelihood function: L( θ ) = ( ) 1 θ 1 1 X X θ θ X ; - = = = n i n n i i i f . ln L( θ ) = ( ) = - + n i i n 1 X ln 1 θ θ ln . ( ) ( ) = + = n i i d d n 1 X θ ˆ θ ˆ L θ ln ln = 0. = = - = - = n i i n i i n n 1 1 X X θ ˆ ln ln .

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HW 8 - STAT 400 Fall 2011 Homework#8(due Friday October 28...

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