# 410Hw09 - STAT 410 Homework #9 (due Friday, November 4, by...

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STAT 410 Homework #9 Fall 2011 (due Friday, November 4, by 3:00 p.m.) 1 - 2. 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) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for θ . b) Determine the Fisher information I ( θ ). 2. Recall: θ ˆ ˆ = ( ) = - - - - n i i n 1 X 1 1 1 ln is an unbiased estimator of θ . Is θ ˆ ˆ an efficient estimator of θ ? If θ ˆ ˆ is not an efficient estimator of θ , find its efficiency. Recall: Y = ( ) = - - n i i 1 X 1 ln has Gamma ( α = n , β = “usual θ ” = 1 θ 1 + ) distribution. 3. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( ) θ 2 θ θ ; x e x x f - = , x > 0, θ > 0. a) Find the sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for θ . b)

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## This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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410Hw09 - STAT 410 Homework #9 (due Friday, November 4, by...

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