Suppose (X1, Y1),..., (Xn, Yn) are i.i.d., with Xi also independent of Yi. Further suppose Xi is normal with mean µ1 and variance 1, and Yi is normal with mean µ2 and variance 1. It is known that µi ≥ 0 for i = 1, 2. The problem is to test the null hypothesis that at most one µi is positive versus the alternative that both µ1 and µ2 are positive.

(i) Determine the likelihood ratio statistic for this problem.

(ii) In order to carry out the test, how would you choose the critical value

(sequence) so that the size of the test is α?