Asymptotic expansions of the inverse of the Beta distribution

Dimitris Askitis

Published 2016-11-11Version 1

In this paper, we shall study the asymptotic behaviour of the $p$-inverse 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 study the monotonicity and limits of $q (a)$, as well as of $\varphi (a) = - a \log q (a)$ and we derive asymptotic expansions of $\varphi$ and $q$ at $0$ and $\infty$. Moreover, we prove some general results on $-inverses, with some interest on their own and 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.