Symbolic Cholesky decomposition of the variance--covariance matrix of the negative multinomial distribution

This note shows a symbolic formula for the square-root-free Cholesky decomposition of the variance--covariance matrix of the negative multinomial distribution. A similar decomposition was given for the multinomial case by Tanabe and Sagae (1984). The evaluation of the symbolic Cholesky factors requires much less arithmetic operations than those with the general Cholesky algorithm. It is applied to obtain a recursive algorithm for generating multivariate normal random numbers which simulate samples from a negative multinomial population, which is similar to Pederson's procedure for sampling from multinomial populations. An explicit formula of a multivariate normal density approximation to negative multinomial distribution is also given.