Uniformly More Powerful, Two-Sided Tests for Hypotheses about Linear Inequalities

Let Xhave a multivariate, p-dimensional normal distribution (p 2) with unknown mean and known, nonsingular covariance . Consider testing H ₀ : b _i 0, for some i = 1,..., k, and b _i 0, for some i = 1,..., k, versus H ₁ : b _i < 0, for all i = 1,..., k, or b _i < 0, for all i = 1,..., k, where b ₁,..., b _k , k 2, are known vectors that define the hypotheses and suppose that for each i = 1,..., k there is an j {1,..., k} (j will depend on i) such that b _i b _j 0. For any 0 < < 1/2. We construct a test that has the same size as the likelihood ratio test (LRT) and is uniformly more powerful than the LRT. The proposed test is an intersection-union test. We apply the result to compare linear regression functions.