Suppose that (Y) is distributed as multivariate normal with unknown covariance matrix and that (N) independent observations are available on (Y). An important special case of the problem studied in this paper is that of testing the null hypothesis that the mean of (Y) is zero against the alternative that it lies in the positive orthant. We propose a statistic (T^{2}) for this testing problem: this closely resembles the wellknown Hotelling's statistic for testing against the unrestricted alternative, and it is also related to some other statistics in the literature. It turns out that our (T^{2}) and the likelihood ratio test (LRT) statistic are equivalent asymptotically but not for linite samples. Some simulations and a comparison of the critical regions of (T^{2}) and LRT in some special cases show that neither of the two can dominate the other uniformly over the parameter space in terms of power. A comparison of the critical regions of (T^{2}) and LRT leads us to conjecture that (T^{2}) is likely to be more (respectively. less) powerful than the LRT when the mean of the multivariate distribution is close to (respectively, away from) the boundary of the parameter space. Computation of the (\rho)-value for (T^{2}) is almost just as straightforward as it is for LRT.
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