Abstract
We assume a random sample of n + 1 from a distribution and study the test statistics for H0 ∑ = ∑0 versus H1 ∑ ≥ ∑0 and H1 ∑ ≥ ∑0 versus H2 ∑ ≠ ∑0, where ∑0 is known. Estimation of ∑ is based on maximizing the likelihood of the location invariant sufficient statistic S, the sample covariance matrix. The one-sided nature of the hypotheses leads to a restricted parameter space and the use of techniques from order restricted inference. The asymptotic distributions of the resulting test statistics are derived and shown to be a poor approximation for small to moderate size samples. An empirical distribution approach is suggested and the power of the tests is discussed.