Continuation. (i) There exists at every significance level a test of H : G = F

Continuation.

(i) There exists at every significance level α a test of H : G = F which has power

> α 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

α = 1

m + n n

.

[(i): let Xi, X i; Yi, Y i (i = 1,..., n) be independently distributed, the X’s with distribution F, the Y ’s with distribution G, and let Vi = 1 if max(Xi, X 1) < min(Yi, Y i )

or max(Yi, Y i ) < min(Xi, X i), and Vi = 0 otherwise. Then Vi has a binomial distribution with the probability p defined in Problem 6.55, and the problem reduces to that of testing p = 1 3 against p > 1 3 .

(ii): Consider the particular alternatives for which P{X < Y } is either 1 or 0.]