Asymptotic expansions of the inverse of the Beta distribution

TL;DR

This paper investigates the asymptotic behavior of the inverse Beta distribution's quantile as a function of its parameters, deriving expansions and algorithms for computation.

Contribution

It introduces new asymptotic expansions for the Beta distribution's quantile and its logarithm, along with relations between special polynomials and computational algorithms.

Findings

Derived asymptotic expansions at 0 and infinity.

Established relations between Bell and N{ o}rlund Polynomials.

Provided algorithms for computing expansion terms.

Abstract

In this work in progress, we study the asymptotic behaviour of the $p$-quantile of the Beta distribution, i.e. the quantity $q$ defined implicitly by $\int_0^q t^{a - 1} (1 - t)^{b - 1} \text{d} t = p B (a, b)$, as a function of the first parameter $a$. In particular, we derive asymptotic expansions of and $q$ and its logarithm at $0$ and $\infty$. Moreover, we provide some relations between Bell and N{\o}rlund Polynomials, a generalisation of Bernoulli numbers. Finally, we provide Maple and Sage algorithms for computing the terms of the asymptotic expansions.