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hw1_stat210a_solutions

hw1_stat210a_solutions - UC Berkeley Department of...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1- Solutions Fall 2006 Issued: Thursday, August 31, 2006 Due: Thursday, September 7, 2006 Problem 1.1 Solution to 1. Let: Y n = 0 , with probability 1 - 1 n n, with probability 1 n Clearly, E ( Y n ) = 1 for all n and, hence, lim n →∞ E ( Y n ) = 1. However, for all δ > 0, 0 P ( | Y n - 0 | ≥ δ ) 1 n and hence, for all δ > 0, lim n →∞ P ( | Y n - 0 | ≥ δ ) = 0 so Y n p 0. 2. Let: Y n = - n, with probability 1 2 n 0 , with probability 1 - 1 n n, with probability 1 2 n Hence E ( Y n ) = 0 and var ( Y n ) = 2 · ( n ) 2 · 1 2 n = 1 for all n and, therefore, lim n →∞ var ( Y n ) = 1. As in item a, for all δ > 0, 0 P ( | Y n - 0 | ≥ δ ) 1 n and hence, for all δ > 0, lim n →∞ P ( | Y n - 0 | ≥ δ ) = 0 so Y n p 0. Problem 1.2 See Examples from section 2.2 in Large Sample Theory, by Erich Lehmann: 1. We have that E ( ¯ X - μ ) 2 = E P n i =1 ( X i - μ ) 2 n 2 . Given independence, E ( ¯ X - μ ) 2 = P n i =1 σ 2 i n 2 0 establishing convergence in quadratic mean ( L 2 -convergence). Convergence in probability follows from convergence in quadratic mean. 1
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2. Once it is proved that the var ( ˜ X n ) var ( ¯ X n ), L 2 convergence of ¯ X n implies L 2 convergence of ˜ X n . Convergence in probability follows. The statement is true in both the “original form” ˜ X = P i X i σ i P i 1 σ i and the “corrected form ˜ X = P i X i σ 2 i P i 1 σ 2 i For the “corrected form”: Write the mean as the estimate for α in the regression model X i = α + ε i . After reweighting by the inverse of the variance, the estimate is
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