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

## TL;DR

This paper investigates efficient computation and inversion methods for the noncentral beta distribution, including stability analysis, asymptotic expansions for large parameters, and initial approximations for solving nonlinear equations.

## Contribution

It introduces stability analysis, asymptotic expansions, and initial approximation techniques for the noncentral beta distribution and its inversion, enhancing computational accuracy and efficiency.

## Key findings

- Stability analysis of recursions for $B_{p,q}(x,y)$
- Asymptotic expansions for large parameters
- Initial approximations for inversion problems

## Abstract

The computation and inversion of the noncentral beta distribution $B_{p,q}(x,y)$ (or the noncentral $F$-distribution, a particular case of $B_{p,q}(x,y)$) play an important role in different applications. In this paper we study the stability of recursions satisfied by $B_{p,q}(x,y)$ and its complementary function and describe asymptotic expansions useful for computing the function when the parameters are large. We also consider the inversion problem of finding $x$ or $y$ when a value of $B_{p,q}(x,y)$ is given. We provide approximations to $x$ and $y$ which can be used as starting values of methods for solving nonlinear equations (such as Newton) if higher accuracy is needed.