Liu, Wei On the existence of tests uniformly more powerful than the likelihood ratio test. (English) Zbl 0812.62020 J. Stat. Plann. Inference 44, No. 1, 19-35 (1995). Summary: For hypotheses concerning linear inequalities and \(k\) normal means, R. L. Berger [J. Am. Stat. Assoc. 84, No. 405, 192-199 (1989; Zbl 0683.62035)] showed that the likelihood ratio test (LRT) is not very powerful, and uniformly more powerful tests which reject in some extra regions can be constructed under some conditions. In this article, we study whether uniformly more powerful tests of the form considered by Berger exist in some situations that are not covered by the results of Berger.First, when the covariance matrix has a special structure which usually arises from the problem of comparing several treatments with a control, we show that there exists no such uniformly more powerful test for \(k=2\), and there always exists such a uniformly more powerful test for \(k\geq 3\). Secondly, when the common variance of the \(k\) independent normal populations is unknown, we again show the nonexistence of such a uniformly more powerful test for \(k=2\), and the existence of such a uniformly more powerful test for \(k\geq 3\). The method of proof can be applied to other situations, though it does not tell how to construct such a uniformly more powerful test. Cited in 2 Documents MSC: 62F03 Parametric hypothesis testing 62H15 Hypothesis testing in multivariate analysis Keywords:comparison with a control; unknown variance; linear inequalities; normal means; likelihood ratio test; uniformly more powerful tests Citations:Zbl 0683.62035 PDFBibTeX XMLCite \textit{W. Liu}, J. Stat. Plann. Inference 44, No. 1, 19--35 (1995; Zbl 0812.62020) Full Text: DOI References: [1] Berger, R. L., Uniformly more powerful tests for hypotheses concerning linear inequalities and normal means, J. Amer. Statist. Assoc., 84, 192-199 (1989) · Zbl 0683.62035 [2] Chow, Y. S.; Teicher, H., Probability Theory (1978), Springer: Springer New York [3] Dunnett, C. W., A multiple comparison procedure for comparing several treatments with a control, J. Amer. Statist. Assoc., 50, 1096-1121 (1955) · Zbl 0066.12603 [4] Gutmann, S., Tests uniformly more powerful than uniformly most powerful monotone tests, J. Statist. Plann. Inference, 17, 279-292 (1987) · Zbl 0635.62021 [5] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1952), Cambridge Univ. Press · Zbl 0047.05302 [6] Marshall, A. W.; Olkin, I., Inequalities: Theory of Majorization and Its Applications (1979), Academic Press: Academic Press New York · Zbl 0437.26007 [7] Sasabuchi, S., A test of a multivariate normal mean with composite hypotheses determined by linear inequalities, Biometrika, 67, 429-439 (1980) · Zbl 0437.62053 [8] Shirley, A. G., Is the minimum of several location parameters positive?, J. Statist Plann. Inference, 31, 67-79 (1992) · Zbl 0748.62013 [9] Stephenson, G., Mathematical Methods for Science Students (1973), Longman: Longman London and New York [10] Tong, Y. L., Probability Inequalities in Multivariate Distributions (1980), Academic Press: Academic Press New York · Zbl 0455.60003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.