AISM 52, 15-27

## 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** have a multivariate,
*p*-dimensional normal distribution (*p* __>__ 2) with unknown
mean *mu* and known, nonsingular covariance
*Sigma*.
Consider testing *H*_{0}:*b*^{'}_{i}
\b*mu* __<__ 0, for some *i*=1,
..., *k*, and *b*^{'}_{i}
\b*mu* __>__ 0, for some *i*=1,..., *k*,
versus *H*_{1}:*b*^{'}_{i}
\b*mu* > 0, for all *i*=1,
..., *k*, or *b*^{'}_{i}
\b*mu* < 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* \in {1,..., *k*} ( *j* will depend on *i*)
such that *b*^{'}_{i}
\b*Sigma* **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.

**Source**
( TeX , DVI )