41. Continuation. (i) There exists at every significance level a a test of H: G = F which has power >a against all continuous alternatives (F,G) with F + G. (ii) There does not exist a nonrandomized unbiased rank test of H against all G+ F at level a-1/(m +"). [(i): let X,, X;; Y,, Y; (i-1,..., n) be independently distributed, the X's with distribution F, the Y's with distribution G, and let V-1 if max(X,, X) < min(Y,, Y) or max(Y, Y) < min(X, X), and V, 0 otherwise. Then V, has a binomial distribution with the probability p defined in Problem 40, and the problem reduces to that of testing p against p > . (ii): Consider the particular alternatives for which P{X < Y) is either 1 or 0.]