79. Consider a one-sided, one-sample, level-a r-test with rejection region t( X) en ' where X = (X\, . .. , Xn ) and t( X) is given by (16). Let an (F) be the rejection probability when X\, ... , Xn are i.i.d. according to a distribution FE§', with §' the class of all distributions with mean zero and finite variance. Then for any fixed n, no matter how large, SUPFEjOan(F) = 1. [Let F be a mixture of two normals, F = yN(I, 02) + (1 - y)N(J.L, 02) with y + (1 - y) J.L = O. By taking y sufficiently close to 1, one can be virtually certain that all n observations are from N(I,0 2 ) . By taking 0 sufficiently small, one can make the power of the r-test against the alternative N(I,0 2) arbitrarily close to 1. The result follows.] Note. This is a special case of results of Bahadur and Savage (1956); for further discussion, see Loh (1985).