An Exact Cholesky Decomposition and the Generalized Inverse of the Variance–Covariance Matrix of the Multinomial Distribution, with Applications
给出了多项分布方差-协方差矩阵的无平方根Cholesky分解的符号公式，计算量远小于通用算法，且不受病态矩阵影响，适用于概率向量元素量级差异大的情况。还给出了Moore-Penrose逆的显式公式和多项分布密度的对称近似表示，有助于统计建模和计算。
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.