Exact probabilities of obtaining estimated non-positive definite between-group covariance matrices

Abstract

The probability that the estimated between-group covariance matrix is not positive definite is computed exactly under certain conditions given in Hill and Thompson (1978) for the balanced single classification multi-vanate analysis of variance with random effects. The computation is done using the Pfaffian method. This method can also be used more generally to calculate the cummulative distribution function of the largest root of the matrix where S₁ and S₂ are independent central Wishart matrices with common covariance matrices. The Pfaffian method is found to be a rapid and highly practical alternative to an algorithm of Venables (1974).

Keywords:

• Largest root distribution
• multivariate analysis of variance
• Pfaffian method
• sample covariance matrices
• genetic covariance

†Presently at: Computer and Statistical Operations, Research and Development Pennwalt Corporation, P.O. Box 1710. Rochester, N w York 14603.

‡Presently at: Rose-Hulman Institute of Technology, Division of Mathematics, 5500 Wabash Ave., Terre Haute, Indiana 47803.

†Presently at: Computer and Statistical Operations, Research and Development Pennwalt Corporation, P.O. Box 1710. Rochester, N w York 14603.

‡Presently at: Rose-Hulman Institute of Technology, Division of Mathematics, 5500 Wabash Ave., Terre Haute, Indiana 47803.

Notes

†Presently at: Computer and Statistical Operations, Research and Development Pennwalt Corporation, P.O. Box 1710. Rochester, N w York 14603.

‡Presently at: Rose-Hulman Institute of Technology, Division of Mathematics, 5500 Wabash Ave., Terre Haute, Indiana 47803.