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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1- Solutions Fall 2007 Issued: Thursday, August 30 Due: Thursday, September 6 Problem 1.1 (a) We have that E ( X- ) 2 / = E h P n i =1 ( X i- ) 2 n 2 i . Given independence, E ( X- ) 2 / = P n i =1 2 i n 2 0 establishing convergence in quadratic mean ( L 2-convergence). Convergence in probability follows from convergence in quadratic mean. (b) 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 stated version X = P i X i i P i 1 i and an alternative version X = P i X i 2 i P i 1 2 i For the form given in the problem: We have var( X ) = n P i 1 i 2 1 n 3 ( i i ) 2 by the harmonic-arithmetic mean inequality. Now, By Jensens inequality, 1 n 3 ( i i ) 2 1 n 2 i 2 i = var ( X n ). For the alternative version: Write the mean as the estimate for in the regression model X i = + i . After reweighting by the inverse of the variance, the estimate is given by = X = P i X i 2 i P i 1 2 i is the best linear unbiased estimate of by the Gauss-Markov theorem and the result follows; 1...
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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