ps7 - Problem Set 7 ECON 837 Prof. Simon Woodcock, Spring...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Problem Set 7 ECON 837 Prof. Simon Woodcock, Spring 2006 Due: Wednesday April 5 1. (Casella and Berger, 1990) Suppose that X 1 ; X 2 ; :::; X n are an iid random sample from a Bernoulli ( p ) distribution. That is, f ( x j p ) = Pr [ X = x j p ] = p x (1 & p ) 1 & x x 2 f ; 1 g ; p 2 [0 ; 1] : Let & x = 1 n P n i =1 x i : (a) Show that & x is an unbiased estimator of p: (b) Show that the variance of & x attains the Cramer-Rao lower bound, so that & x is the best unbiased estimator of p: 2. [Final Exam, 2004] Let y = & i + " where i is an n-vector of ones and " N ( ; I ) : (a) What is the MLE of & ? What is its asymptotic distribution? (b) Derive the likelihood ratio statistic (and give its asymptotic distribution) for test- ing H : & = 0 ; H 1 : & 6 = 0 : (c) Derive the Wald test statistic (and give its asymptotic distribution) for testing the same hypothesis. (d) Derive the score (LM) test statistic (and give its asymptotic distribution) for testing the same hypothesis. (e) Suppose you are testing &at the 0.05 level.Sketch the general shape of the func- tion Pr [ reject H j & ] as a function of & . Indicate how the shape of this function changes as the sample size increases. 3. [Final Exam, 2005] Suppose you are interested in the regression model y = X & + " where y is N 1 ; X is N k; & is k 1 ; and " is N 1 : Assume that " N (0 ; 2 I n ) , that y...
View Full Document

This note was uploaded on 10/08/2011 for the course ECON 837 taught by Professor Thewalt during the Spring '09 term at Simon Fraser.

Page1 / 3

ps7 - Problem Set 7 ECON 837 Prof. Simon Woodcock, Spring...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online