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?