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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 2 Fall 2006 Issued: Thursday, September 7, 2006 Due: Thursday, September 14, 2006 Graded problems Problem 2.1 Suppose that X i , i = 1 , . . . , n are i.i.d. Poisson random variables with parameter . Show that T = n i =1 X i is sufficient in two ways: (a) first use direct methods: compute the conditional distribution given T = t . (b) apply the factorization theorem. Problem 2.2 Let X 1 , . . . , X n be i.i.d. samples from the uniform Uni[ , ] distribution, where and are unknown. Suppose that we wish to estimate the mean = + 2 under the quadratic loss L (( , ) , a ) = ( - a ) 2 . The sample mean X is one reasonable estimate of . Show that the estimator ( X 1 , . . . , X n ) : = E X | min X i , max X i improves upon X , and moreover that ( X 1 , . . . , X n ) = min X i +max X i 2 ....
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