ST202 Probability Distribution Theory and Inference Lecturer: Kostas Kalogeropou

ST202 Probability Distribution Theory and Inference Lecturer: Kostas Kalogeropoulos (Oce B610) Lent Term 2011 Week 8: Exercises 1. Suppose that we have a random sample Y1; : : : ; Yn from a N(0; 2). (a) Find the uniformly most powerful test for the testing problem H0 : 2 = 20 versus H1 : 2 = 21 0 20? Hint: Check whether the rejection region of the previous test depends on 21.
ST202 Probability Distribution Theory and Inference Lecturer: Kostas Kalogeropoulos (Oce B610) Lent Term 2011 Week 8: Exercises 1. Suppose that we have a random sample Y1; : : : ; Yn from a N(0; 2). (a) Find the uniformly most powerful test for the testing problem H0 : 2 = 20 versus H1 : 2 = 21 0 20? Hint: Check whether the rejection region of the previous test depends on 21. (c) Is this a uniformly most powerful test for the testing problem H0 : 2 20 versus H1 : 2 > 20? Hint: Show that sup 220 P2(Y 2 R) = P20 (Y 2 R) where R is the rejection region of the previous test. 2. Let Y = (Y1; Y2; : : : ; Yn) be a random sample from an Exponential() distribu- tion. Obtain the likelihood ratio test statistic for the test H0 : = 0 versus 6= 0. Can we construct a test based on this statistic? 3. Suppose that X1; :::;Xn and Y1; :::; Yn are two independent random samples from two Normal distributions with mean 1 and 2 respectively. Assume that both distributions have variance 1. (a) Find the maximum likelihood estimator for 1 2 and for 1 when 1 = 2 = . (b) Find the likelihood ratio test statistic T for H0 : 1 = 2 against H1 : 1 6= 2. (c) Find the distribution of 2 log T under H0 and construct a size test.
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