16. Testing for Independence in a Bivariate Normal Population Distribution. Let (Xi, Yi), i 1,...,n be

16. Testing for Independence in a Bivariate Normal Population Distribution. Let (Xi, Yi), i ¼ 1,...,n be a random sample from some bivariate normal population distribution N(m, S) where s2 1 ¼ s2 2. In this case, we know that the bivariate random variables are independent iff the correlation between them is zero. Consider testing the hypothesis of independence H0: r ¼ 0 versus the alternative hypothesis of dependence Ha: r 6¼ 0.

(a) Define a size .05 GLR test of the independence of the two random variables. You may use the limiting distribution of the GLR statistic to define the test.

(b) In a sample of 50 observations, it was found that s2 x

¼ 5.37, s2 y ¼ 3.62, and sxy ¼ .98. Is this sufficient to reject the hypothesis of independence based on the asymptotically valid test above?

(c) Show that you can transform the GLR test into a test involving a critical region for the test statistic w ¼ sxy= s2 x þ s2 y

 =2 h i.

(d) Derive the sampling distribution of the test statistic W defined in

c) under H0. Can you define a size .05 critical region for the test statistic? If so , test the hypothesis using the exact (as opposed to asymptotic)

size .05 GLR test.