Mathematical Statistics and Data Analysis - Homework 4 | MATH 4620, Assignments of Mathematical Statistics

Download Mathematical Statistics and Data Analysis - Homework 4 | MATH 4620 and more Assignments Mathematical Statistics in PDF only on Docsity! Math 4620 Spring 2009 Homework 4 Jason Stover These problems were copied from J. Rice’s book, Mathematical Statistics and Data Anal- ysis, second edition, chapter 9. 1. A coin is thrown independently 10 times to test the hypothesis that the probability of heads is 1/2 versus the alternative that the probability is not 1/2. The test rejects if either 0 or 10 heads are observed. (a) What is the significance level of this test? (b) If in fact the probability of heads is 0.1, what is the power of the test? 2. Let X have one of the following distributions: X H0 HA x1 0.2 0.1 x2 0.3 0.4 x3 0.3 0.1 x4 0.2 0.4 (a) Compare the likelihood ratio, Λ, for each possible value X and sort the xi according to Λ. (b) What is the likelihood ratio test of H0 versus HA at level α = 0.2? What is the test at level α = 0.5? 3. Let X1, ..., Xn be a sample from a Poisson distribution. Find the likelihood ratio for testing H0 : λ = λ0 versus HA : λ = λ0, where λ1 > λ0. Use the fact that the sum of independent Poisson random variables follows a Poisson distribution to explain how to determine a rejection region for a test at level α. 4. Show that the test of the previous problem is uniformly most powerful for testing H0 : λ = λ0 versus HA : λ > λ0. 5. Suppose that X1, ..., Xn form a random sample from a density function, f(x|θ), for which T is a sufficient statistic for θ. Show that the likelihood ratio test of H0 : θ = θ0 versus HA : θ = θ1 is a function of T . Explain how, if the distribution of T is known under H0, the rejection region of the test may be chosen so that the test has the level α.