Powers of some one-sided multivariate tests with the population covariance matrix known up to a multiplicative constant

Abstract

For a multivariate normal population, Kudo (Biometrika 50 (1963) 403) and Shorack (Ann. Math. Statist. 38 (1967) 1740) derived the likelihood ratio tests of the null hypothesis that the mean vector is zero with a one-sided alternative for a known covariance matrix and for a covariance matrix which is known up to a multiplicative constant, respectively. Because these tests may be tedious to use, Tang et al. (Biometrika 76 (1989) 577) developed an approximate likelihood ratio test and Follmann (J. Amer. Statist. Assoc. 91 (1996) 854) proposed a one-sided modification of the usual chi-squared test for an unordered alternative. We consider a modification of Follmann's test which performs better than Follmann's test at some alternatives, and we derive expressions for the powers of the new test and Follmann's test for the cases considered here. For multivariate normal distributions with dimension no more than three and known covariance matrix, we consider the power functions of Kudo's test and the Tang–Gnecco–Geller test. Using these exact results and Monte-Carlo simulations, we study the powers of these one-sided tests.