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410Hw06 - STAT 410 Fall 2011 Homework#6(due Friday October...

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STAT 410 Fall 2011 Homework #6 (due Friday, October 14, by 3:00 p.m.) 1 - 4. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( ) ( ) ( ) θ 1 1 θ θ ; X x x f - + = , 0 < x < 1, θ > – 1. 1. a) Obtain the method of moments estimator of θ , θ ~ . b) Obtain the maximum likelihood estimator of θ , θ ˆ . 2. a) What is the probability distribution of W = – ln ( 1 – X ) ? b) What is the probability distribution of Y = ( ) = - - n i i 1 X 1 ln = = n i i 1 W ? c) Is the maximum likelihood estimator of θ , θ ˆ , an unbiased estimator of θ ? If θ ˆ is not an unbiased estimator of θ , construct an unbiased estimator θ ˆ ˆ of θ based on θ ˆ . 3. For an estimator θ ˆ of θ , define the Mean Squared Error of θ ˆ by MSE ( θ ˆ ) = E [ ( θ ˆ θ ) 2 ] . We showed in class that MSE ( θ ˆ ) = E [ ( θ ˆ θ ) 2 ] = ( E ( θ ˆ ) θ ) 2 + Var ( θ ˆ ) = ( bias ( θ ˆ ) ) 2 + Var ( θ ˆ ) . Find MSE ( θ ˆ ) for the maximum likelihood estimator of θ . 4. Consider θ ˆ = ( ) ( ) = = - - - - n n i i i i n n 1 2 1 2 X 1 X 1 3 .
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