A Wiener Germ approximation of the noncentral chi square distribution and of its quantiles

Summary

The cumulative distribution function (cdf) of the noncentral χ² distribution with positive degrees of freedom ν > 0 and a noncentrality parameter δ² ≥ 0 is usually expressed as an infinite weighted sum of central χ² cdf’s. For the purpose of numerical evaluation this infinite sum is being approximated by a finite sum. For large values of the noncentrality parameter, the sum converges slowly. Alternative approximation algorithms have been proposed instead in the literature. A comparison of these is given in Johnson & Kotz (1970). Most of the approximation algorithms have advantages for certain values of the arguments/parameters and perform poorly for other values. We are proposing an approximation algorithm that has a very solid theoretical background and is surprisingly accurate for extremely large set of arguments/parameter values. It is also applied for a reliable approximation of the quantiles of the distribution for large values of noncentrality and degrees of freedom. Although being asymptotic in spirit (with respect to degrees of freedom ν), the algorithm gives quite accurate approximation even down to ν = 1.