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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7 Fall 2007 Issued: Thursday, October 25 Due: Thursday, October 18 Reading: Keener: Chapter 10, 11. B & D: Chapter 5. Problem 7.1 Suppose that X 1 ,...,X n are i.i.d. N (0 , 2 ). (a) Show that ( X ) = C n n i =1 | X i | is a consistent estimator of if and only if C = p / 2. (b) Show that the MLE of is given by MLE ( X ) = q 1 n n i =1 X 2 i . Determine the asymp- totic relative efficiency of with c = p / 2 compared to the MLE MLE . Problem 7.2 Suppose that X 1 ,...X n are i.i.d from the normal location model N ( , 1), and that we wish to estimate the critical or cutoff value g a ( ) = P [ X 1 a ], where a R is some fixed number. (a) Let denote the CDF of the standard normal distribution. Show that the estimator n ( X ) = r n n- 1 ( a- X n ) is an unbiased estimator of g a ( ). Prove that n ( n- g a ( )) d N (0 ,...
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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