Suppose that we have n independent observations x_1, ..., x_n from a normal dist

Question

Suppose that we have n independent observations x_1, ..., x_n from a normal distribution with mean mu and variance sigma^2, and we want to test H_0:sigma^2 = 1 H_1: sigma^2 > 1 Kind the maximum likelihood estimator of mu when the null hypothesis is true. Calculate the Likelihood Ratio Test Statistic T = 2 log (max_mu, sigma^2 L (mu, sigma^2)) - 2 log (max_mu L (mu, 1)) Explain as clearly as you can what happens to T when our estimate of sigma^2 is less than 1. Show that the GLRT test is the same as a test which rejects the null hypothesis when s^2 > k.

Explanation / Answer

a. when the variance is equal to 1 the maximum likelihood estimator for papulaton mean is sample mean (x-bar)

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