One-sided test of a covariance matrix with a known null value

We assume a random sample of n + 1 from a MVNK(μ,∑) distribution and study the test statistics for H₀ : ∑= ∑₀ versus Hi : E ≥ E₀ and H1 : ∑≥ ∑₀ versus H₂ : ∑ ≠∑0, where ∑0 is known. Estimation of ∑ is based on maximizing the likelihood of the location invariant sufficient statistic 5, 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.