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February, 1995 Uniformly More Powerful, One-Sided Tests for Hypotheses About Linear Inequalities
Huimei Liu, Roger L. Berger
Ann. Statist. 23(1): 55-72 (February, 1995). DOI: 10.1214/aos/1176324455


Let $\mathbf{X}$ have a multivariate, $p$-dimensional normal distribution $(p \geq 2)$ with unknown mean $\mathbf{\mu}$ and known, nonsingular covariance $\mathbf{\Sigma}$. Consider testing $H_0: \mathbf{b}'_i\mathbf{\mu} \leq 0$, for some $i = 1, \ldots, k$, versus $H_1: \mathbf{b}'_i\mathbf{\mu} > 0$, for all $i = 1, \ldots, k$, where $\mathbf{b}_1, \ldots, \mathbf{b}_k, k \geq 2$, are known vectors that define the hypotheses. For any $0 < \alpha < 1/2$, we construct a size-$\alpha$ test that is uniformly more powerful than the size-$\alpha$ likelihood ratio test (LRT). The proposed test is an intersection-union test. Other authors have presented uniformly more powerful tests under restrictions on the covariance matrix and on the hypothesis being tested. Our new test is uniformly more powerful than the LRT for all known nonsingular covariance matrices and all hypotheses. So our results show that, in a very general class of problems, the LRT can be uniformly dominated.


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Huimei Liu. Roger L. Berger. "Uniformly More Powerful, One-Sided Tests for Hypotheses About Linear Inequalities." Ann. Statist. 23 (1) 55 - 72, February, 1995.


Published: February, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0821.62011
MathSciNet: MR1331656
Digital Object Identifier: 10.1214/aos/1176324455

Primary: 62F03
Secondary: 62F04 , 62F30 , 62H15

Keywords: acute cone , Intersection-union test , likelihood ratio test , linear inequalities hypotheses , obtuse cone , Uniformly more powerful test

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 1 • February, 1995
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