Answer Happy

PROBLEM 2. The simple Gaussian regression model is given by (7.1) Y; = a + Britei, i = 1, ...,,n, where the dependent variables, {Y;} are stochastic independent, normal distributed with constant variance o?. The explanatory variables, {li}, are fixed and known. We further assume that Szı = 2–1 (1; — 7)2 is strictly positive. The main subject here is show that the two-sided t-test for a hypothesis about the regression coefficient is a LRT.

n/2 1 = a) Show that the simultaneous density for {Y;} can be written as f(yla,B, 02) = (21)-n/2 G" exp{262Q{a,b)} (7.2) Q(a,b) del (yi - -1 – I;B)2 n i=1 Define (7.3) â=Y - BT, Ŝ= Sym Sza ô2 = n-'Q(@B). n b) Verify that (7.4) Q(a,B) = Q(@,B) +n{(a – a) + Fn(B – B)}" + Szz (B – B)?. c) Prove by (7.4) that the estimators defined by (7.3) are the MLE's. Consider the hypothesis testing problem for the regression coefficient given by Ho: B = Bo versus H :B + Bo. with a fixed significance level y. Let (a, b, 72) be the MLE under Hy so that the LRT statistic is f(Y|ā, ß,72) f(Yla, , ô2) A= — d) Show by (7.2) and (7.4) that õ2 = n-'Qlā, ). e) Deduce from the last point that A = (@2/72)n/2. f) Prove with help of (7.4) that Qlā, B) = QQ,B) + (B – Bo) Sqr. Let n(B – Bo)? t2 02 SCO with ôz = nô2/(n − 2), a bias corrected version of ô2. g) Conclude that the LRT is equivalent to the two-sided t-test that reject iff t? > fy,1,n-2. The critical value is the 7- quantile in the F1,n-2-distribution. h) Suppose that the explanatory variables are iid and normal distributed independent of the noise variables. Is the test you have developed still admissable? -