Abstract | Consider testing H0: μ1=⋯=μp=0 versus the multivariate one-sided alternative H1: μ1⩾0,…,μp⩾0 based on a sample from a p-dimensional normal distribution Np(μ,Σ) with Σ unknown, where μ=(μ1,…,μp). Perlman (Ann. Math. Statist. 40 (1969) 549) obtained the likelihood ratio test (LRT) statistic and its null distribution, while Shorack (Ann. Math. Statist. 38 (1967) 1740) and Silvapulle (J. Multivariate Anal. 55 (1995) 312) studied an alternative test statistic. Neither test is similar, however, hence both are biased, and neither dominates the other in terms of power. Tang (J. Am. Statist. Assoc. 89 (1994) 1006) proposed a test that is similar, unbiased, and everywhere more powerful than the original LRT. Here we study a new class of conditional tests based on the LRT statistic and obtain the conditional test that is most nearly similar within this class. The resulting test is more powerful than the LRT for most alternatives, is not dominated in power by Tang's test, and is more convenient to apply. Furthermore, unlike Tang's test, the conditional test always accepts H0 when μ̂≡(μ̂1,…,μ̂p)=(0,…,0), where μ̂ is the MLE of μ under H1. Similar results are found for a conditional version of the Shorack–Silvapulle test. |