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Quiz3Aans - θ Show work L θ = ∏ =-n i i x 1& 1&...

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STAT 410 Spring 2008 Version A Name ANSWERS . Quiz 3 (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 of size n from the distribution with probability density function ( ) ( ) ( ) ° X X 1 ° ° ; x x f x f - = = , x > 1. a) (5) Ω = { θ > 1 } . Find the maximum likelihood estimator ° ˆ of θ . Show work. L( θ ) = ( )
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Unformatted text preview: θ . Show work. L( θ ) = ( ) ∏ =-n i i x 1 & 1 & . ln L( θ ) = ( ) & = ⋅--n i i x n 1 ln & 1 & ln . ( ) & =--= n i i d d x n 1 ln 1 & & & L ln = 0. & = + = n i i x n 1 ln 1 & ˆ . b) (5) Ω = { θ > 2 }. Find the method of moments estimator & ~ of θ . Show work. E(X) = ( ) ( ) 2 & 1 & 1 & 1 & X--=-= ± ± ∞ ∞ ∞-⋅ ⋅ dx x x dx x f x . 2 & 1 & 1 1--= = & = ⋅ x x n n i i . 1 1 2 & ~--= x x ....
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