## The Annals of Statistics

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

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

Huimei Liu and Roger L. Berger

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324455

**Digital Object Identifier**

doi:10.1214/aos/1176324455

**Mathematical Reviews number (MathSciNet)**

MR1331656

**Zentralblatt MATH identifier**

0821.62011

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F03: Hypothesis testing

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176324455