Let X1, , Xn be independent normal variables with variance 1 and means 1, , n, and

Let X1, …, Xn be independent normal variables with variance 1 and means ξ1, …, ξn, and consider the problem of testing H : ξ1 =···= ξn = 0 against the alternatives K = {K1,..., Kn}, where Ki : ξ j = 0 for j = i, ξi = ξ (known and positive). Show that the problem remains invariant under permutation of the X’s and that there exists a UMP invariant test φ0 which rejects when eξ Xi > C, by the following two methods.

(i) The order statistics X(1) < ··· < X(n) constitute a maximal invariant.

(ii) Let f0 and fi denote the densities under H and Ki respectively. Then the level-α
test φ0 of H versus K : f = (1/n)
fi is UMP invariant for testing H versus K.
[(ii): If φ0 is not UMP invariant for H versus K, there exists an invariant test φ1 whose (constant) power against K exceeds that of φ0. Then φ1 is also more powerful against K
.]