picture attached

Exercise 16.7.2. Suppose X1, ... . X, are iid N(ux, '), and Y. ....Ym are iid N(Hy, "), and the X/'s and Y's are independent. We wish to test wx = wy, with of unknown. Then the hypotheses are Ho : HX = MY, > 0 versus HA : HX / HY,' >0. (16.102) (a) Find the MLEs of the parameters under the null hypothesis. (b) Find the MLEs of the parameters under the alternative hypothesis. (c) Find the LRT statistic. (d) Letting and of be the MLEs for of under the null and alternative, respectively, show that (n + m) (0 -81) = KM,m (X - 7)? (16.103) 278 Chapter 16. Likelihood Testing and Model Selection for some constant kam that depends on n and m. Find the ke,m. (e) Let T be the two-sample /-statistic, T = X - V (16.104) S pooled Vit m where sois the pooled variance estimate, Spooled [(x - x)2 + [(yi - y)2 n + m -2 (16.105) What is the distribution of T under the null hypothesis? (You don't have to prove it; just say what the distribution is.) (f) Show that the LRT statistic is an increasing function of T2