AISM 52, 15-27

Uniformly more powerful, two-sided tests for hypotheses about linear inequalities

Huimei Liu

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

Abstract.    Let X have a multivariate, p-dimensional normal distribution (p > 2) with unknown mean mu and known, nonsingular covariance Sigma. Consider testing H₀:b^'_i \bmu < 0, for some i=1, ..., k, and b^'_i \bmu > 0, for some i=1,..., k, versus H₁:b^'_i \bmu > 0, for all i=1, ..., k, or b^'_i \bmu < 0, for all i=1, ..., k, where b₁,..., b_k, k > 2, are known vectors that define the hypotheses and suppose that for each i=1,..., k there is an j \in {1,..., k} ( j will depend on i) such that b^'_i \bSigma b_j < 0. For any 0 < alpha < 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.

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