A Method for the Evaluation of Cumulative Probabilities of Bivariate Distributions using the Pearson Family
A technique is developed for the approximation of bivariate cumulative distribution function values through the use of the bivariate Pearson family when moments of the distribution to be approximated are known. The method utilizes a factorization of the joint density function into the product of a marginal density function and an associated conditional density, permitting the expression of the double integral in a form amenable to the use of specialized Gaussian-type quadrature techniques for numerical evaluation of cumulative probabilities. Such an approach requires moments of truncated Pearson distributions, for which a recurrence relation is presented, and moments of the conditional distributions for which quartic expressions are used. It is shown that results are of high precision when this technique is employed to evaluate cumulative distribution functions that are of the Pearson class.
Key WordsBivariate distribution Pearson system evaluation of distribution functions quadrature
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