Let X1, ..., Xn be independent normal with means 1, ..., n and variance 1. (i) Apply

Let X1, ..., Xn be independent normal with means θ1, ..., θn and variance 1.

(i) Apply the results of the preceding problem to the testing of H : θ1 = ··· =

θn = 0 against K :

θ2 i = r2, for any fixed r > 0.

(ii) Show that the results of (i) remain valid if H and K are replaced by H
:

θ2 i ≤ r2 0, K
:

θ2 i ≥ r2 1 (r0 < r1).