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

Abstract

Let Xhave a multivariate, p-dimensional normal distribution (p ≥ 2) with unknown mean μ and known, nonsingular covariance Σ. Consider testing H ₀ : b _iμ ≤ 0, for some i = 1,..., k, and b _iμ ≥ 0, for some i = 1,..., k, versus H ₁ : b _i μ < 0, for all i = 1,..., k, or b _i ^μ < 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 < α < 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.