GammaCHI: A package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

Abstract

A Fortran 90 module GammaCHI for computing and inverting the gamma and chi-square cumulative distribution functions (central and noncentral) is presented. The main novelty of this package are the reliable and accurate inversion routines for the noncentral cumulative distribution functions. Additionally, the package also provides routines for computing the gamma function, the error function and other functions related to the gamma function. The module includes the routines cdfgamC, invcdfgamC, cdfgamNC, invcdfgamNC, errorfunction, inverfc, gamma, loggam, gamstar and quotgamm for the computation of the central gamma distribution function (and its complementary function), the inversion of the central gamma distribution function, the computation of the noncentral gamma distribution function (and its complementary function), the inversion of the noncentral gamma distribution function, the computation of the error function and its complementary function, the inversion of the complementary error function, the computation of: the gamma function, the logarithm of the gamma function, the regulated gamma function and the ratio of two gamma functions, respectively. The computation and inversion of gamma and chi-square cumulative distribution functions (central and noncentral) as well as the computation of the error and gamma functions is needed in many problems of applied and mathematical physics. PROGRAM SUMMARY Solution method:The algorithms use different methods of computation depending on the range of parameters: asymptotic expansions, quadrature methods, etc. Restrictions:In the inversion of the central gamma/chi-square distribution functions, very

Sharp bounds for cumulative distribution functions

“…The bounds (9) and (10) can be compared with the results is [3] (eqs. (8) and (17)). For instance, Eq.…”

“…Additional bounds can be obtained from the recurrences alone (8), without invoking Theorem 1. Indeed, because the P and the Q functions are positive P µ (x, y) > g µ+1 (x, y) and Q µ (x, y) > g µ (x, y).…”

On the computation and inversion of the cumulative noncentral beta distribution function

“…In [4], [1] and [2] the definition of the noncentral beta function is 6) and the relation with our definition is…”

“…For the inversion with respect to y, we have the equation This corresponds with finding the quantiles of the distribution. We try to find an approximation of x or y by using the representation in (6.30) and by inverting first the complementary error function, that is by A simple and efficient algorithm for computing the inverse of the complementary error function is included, for example, in the package described in [6]. Hence, we consider the numerical inversion of (7.3) as a known problem and concentrate on finding from the computed value ζ 0 the requested value of x or y.…”

Efficient algorithms for the inversion of the cumulative central beta distribution

“…The function Γ * (x) is computed using the function gamstar included in a previous package developed by the authors [8].…”