UNIFORMLY MORE POWERFUL, TWO-SIDED TESTS FOR HYPOTHESES ABOUT LINEAR INEQUALITIES

HUIMEI LIU

Department of Statistics, National Chung Hsing University, 67 Ming Sheng East Rd., Sec. 3, Taipei 10433, Taiwan, R.O.C.

(Received July 15, 1996; revised September 1, 1998)

Abstract. Let X X X X X X X X have a multivariate, p-dimensional normal distribution (p ≥ 2) with unknown mean μ μ μ μ μ μ μ μ and known, nonsingular covariance Σ Σ Σ Σ Σ Σ Σ Σ. Consider testing H_{0} : b′_{i}μ μ μ μ μ μ μ μ ≤ 0, for some i = 1, …, k, and b′_{i}μ μ μ μ μ μ μ μ ≥ 0, for some i = 1, …, k, versus H_{1} : b′_{i}μ μ μ μ μ μ μ μ > 0, for all i = 1, …, k, or b′_{i}μ μ μ μ μ μ μ 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}Σ Σ Σ Σ Σ Σ Σ Σb_{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.

Key words and phrases: Intersection-union test, likelihood ratio test, linear inequalities hypotheses, uniformly more powerful test.