The Annals of Statistics

Uniformly More Powerful, One-Sided Tests for Hypotheses About Linear Inequalities

Huimei Liu and Roger L. Berger

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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.

Article information

Ann. Statist., Volume 23, Number 1 (1995), 55-72.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F03: Hypothesis testing
Secondary: 62F04 62F30: Inference under constraints 62H15: Hypothesis testing

Uniformly more powerful test linear inequalities hypotheses likelihood ratio test acute cone obtuse cone intersection-union test


Liu, Huimei; Berger, Roger L. Uniformly More Powerful, One-Sided Tests for Hypotheses About Linear Inequalities. Ann. Statist. 23 (1995), no. 1, 55--72. doi:10.1214/aos/1176324455.

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