411_hw07_soln

411_hw07_soln - Stat 411 Homework 07 Solutions 1. The...

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Unformatted text preview: Stat 411 Homework 07 Solutions 1. The distribution N ( , 1) is a regular one-parameter exponential family problem with K ( x ) = x . Therefore, T = n i =1 X i is a complete sufficient statistic for and, consequently, the MVUE of is X = T/n . It is easy to check that = X 2- 1 /n is an unbiased estimator of = 2 . Since its a function of the complete sufficient statistic, by the LehmannScheffe theorem, it must be the MVUE. 2. Problem 7.5.3 from the text . (a) The beta distribution with PDF f ( x ) = x - 1 , with x (0 , 1) and > 0, is a regular exponential family. That is, f ( x ) = exp { log + ( - 1) log x } . Since K ( x ) = log x here, a complete sufficient statistic, based on X 1 ,...,X n iid f ( x ), is T = n i =1 log X i . So too is T = exp { T /n } = ( X 1 X 2 X n ) 1 /n , the geometric mean, since the function t 7 e t/n is one-to-one. (b) The log-likelihood function ( ) = n i =1 log f ( X i ) is ( ) = n log + ( - 1) n X i =1 log X i . Differentiating and setting equal to zero gives the likelihood equation: n + n X i =1 log X i = 0 . Therefore, the MLE is =- n/ n i =1 log X i =- 1 / log T , a function of the geometric mean T from part (a)....
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This note was uploaded on 03/12/2012 for the course STAT 411 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

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411_hw07_soln - Stat 411 Homework 07 Solutions 1. The...

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