Nonnegative Cholesky Decomposition and Its Application to Association of Random Variables

Abstract

The concept of a multivariate family of distributions indexed by a covariance scale parameter $\Sigma $ is formally defined and examples given. The multivariate normal is one such family. Sufficient conditions are given so that a positive definite matrix has a nonnegative Cholesky decomposition. These conditions also yield the association of random variables with a covariance scale parameter distribution. These results are related to other matrix results and to Barlow and Proschan’s stronger conditions (Statistical Theory of Reliability and Life Testing; Probability Models, Holt, Rinehart, Winston, New York, 1975), for the association of the multivariate normal, namely, $\lambda _{ij} \leqq 0,i \ne j$ where $\Sigma ^{ - 1} = \Lambda = \{ \lambda _{ij} \}$.