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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Midterm ExaminationSolutions Fall 2006 Problem 1.1 [18 points total] Suppose that X i , i = 1 ,...,n are i.i.d. samples from the uniform Uni[0 , ] distribution. (a) Find a onedimensional sufficient statistic for estimating . (b) Compute the maximum likelihood estimate MLE based on ( X 1 ,...,X n ). Using an elementary argument, show that MLE p * as n + . (c) Consider the estimator of given by ( X ) = 2 n n i =1 X i . Is it unbiased? Is it admissible under squared error loss? Justify your answers. Now suppose that we view the parameter as a random variable ~ , and assume a Pareto prior density of the form ( ) =   1 I [ ] , for all > , where > 0 and > 2 are fixed. (d) Compute the prior mean of the random variable ~ . (e) Compute the posterior distribution of ~ conditioned on X = ( X 1 ,...,X n ). (f) Compute the Bayes estimate of ~ under quadratic loss. Hint: New calculation may not be required given previous parts to the question. Solution 1.1: (a) By independence, we have p ( x ; ) = n Y i =1 1 I [ x i ] for x i =  n I [max( x i ) ] , so that Z = max { X 1 ,...,X n } is sufficient by the factorization criterion. (b) From part (a), the log likelihood takes the form L ( ) = n log( ) for max { X i } , and otherwise, so that the MLE is given by MLE = max { X 1 ,...,X n } . For any (0 , ), we compute P [  max X i  > ] = n Y i =1 P [ X i  ] = h 1 i n as n + , so that consistency of the MLE follows. 1 (c) We compute E [ ( X )] = 2 n n X i =1 E [ X i ] = 2 n ( n 2 ) = , so that the estimator is unbiased. However, since Z = max { X i } is sufficient from part (a) and this estimator depends on other information, the RaoBlackwell theorem dic tates that we can construct a better estimator ( X ) = E [ ( X )  Z ]. The strict convexity of quadratic loss ensures that will dominate , so that must be inadmissible....
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This note was uploaded on 10/17/2009 for the course STAT 210a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.
 Fall '08
 Staff
 Statistics

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