stat210a_fall07_hw1

# stat210a_fall07_hw1 - UC Berkeley Department of Statistics...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1 Fall 2007 Issued: Thursday, August 30 Due: Thursday, September 6 Notation: The symbols { d → , p → , m.s. → } denote convergence in distribution, probability and mean-square respectively. Problem 1.1 Suppose X i is a sequence of independent random variables with a common mean E ( X i ) = μ for all i and different variances E ( X i − μ ) 2 = σ 2 i . (a) Show that ¯ X n = P n i =1 X i n m.s. → μ as long as ∑ n i =1 σ 2 i = o ( n 2 ). Under the same assump- tions, does ¯ X n p → μ ? Why or why not? (b) Now consider the estimate tildewide X n = P n i =1 1 σ i X i P n i =1 1 σ i . Prove that var( tildewide X n ) ≤ var( ¯ X n ). Use this to conclude that, under the same conditions as in part (a), tildewide X n p → μ . Problem 1.2 Consider an i.i.d. sample { X 1 ,...,X n } from the uniform distribution on [0 ,θ ], and the estimator M n = max { X 1 ,...,X n } ....
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stat210a_fall07_hw1 - UC Berkeley Department of Statistics...

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