An Exact Cholesky Decomposition and the Generalized Inverse of the Variance–Covariance Matrix of the Multinomial Distribution, with Applications

Abstract

SUMMARY A symbolic formula is given for the square-root-free Cholesky decomposition of the variance–covariance matrix of the multinomial distribution. The evaluation of the symbolic Cholesky factors requires much fewer arithmetic operations than does the general Cholesky algorithm. Since the symbolic formula is not affected by an ill-conditioned matrix, it is particularly useful when the elements of a probability vector are of quite different orders of magnitude. A simpler formula is obtained for Pederson's procedure of sampling from a multinomial population. An explicit formula of the Moore–Penrose inverse of the variance–covariance matrix is given as well as a symmetric representation of a multinomial density approximation to the multinomial distribution. These formulae facilitate symmetric manipulation of the matrix and are useful in statistical modelling and computation involving the logistic density transformation of the multinomial distribution and in computer simulations of dynamic models in population genetics. Each element of the Cholesky factors is given interesting probabilistic interpretations.