Characteristic values and triangular factorization of the covariance matrix for multinomial, dirichlet and multivariate hypergeometric distributions and some related results

Abstract

The three distributions mentioned in the title have identical covariance structure. The (k×k) covariance matrix is uniquely defined (up to a scale factor) by a set of k positive parameters and its characteristic values are separated by these parameters as shown in the paper. This result is used to obtain a lower bound of the (relative) efficiency of least squares in terms of the parameters when the covariance matrix of the error terms is of the structure as considered here. Some other results such as the triangular factorization of the matrix and the Moore-Penrose inverse of the ‘extended’ matrix are also given.