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

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.

Keywords:

• Order restricted inference
• restricted maximum likelihood estimation
• Chi-bar square distribution
• quality control
• variance components