Statistical Densities, Cumulatives, Quantiles, and Power Obtained by S-System Differential Equations

Abstract

A comprehensive approach for the analysis of statistical univariate distributions is advanced. The method is based on recasting a distribution in a canonical set of first-order nonlinear differential equations called an S system, in which the derivative of each variable equals a difference of products of power-law functions. The recasting techniques are illustrated with some examples from central and noncentral probability distributions. The density and the cumulative, because of their close relationship, can be simultaneously evaluated within one S system. A simple inversion procedure permits the calculation of quantiles. When, in addition, a noncentral distribution and its central counterpart are represented in one S-system, the power function and the inverse power function are obtainable. Given a set of initial conditions, any differential equation solver can generate the desired solution. Particularly convenient and efficient is the program ESSYNS, designed specifically to solve and analyze systems of differential equations in the canonical S-system form. The proposed methods are very general and applicable to multiparameter distributions. In this article, results are presented for the normal and the central and noncentral chi-squared and F distributions.

Key Words:

• Distributions
• Statistical computing