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 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,..., bk, 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 bj < 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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