Evaluation and approximation of multivariate cumulative joint probabilities

Abstract

A method for approximation of cumulative joint probabilities is presented in its multivariate generalization. The technique utilizes joint moments, through fourth order, of the distribution to be approximated to determine Pearson-type differential equation constants and associated conditional moments. Nonstandard Gaussian quadrature techniques are employed for integral evaluation, yielding an expression involving only univariate Pearson distribution function values. Under stated conditions, this method produces mathematically exact evaluations of distribution functions that are of the Pearson class. Illustrations are given for the trivariate case.