Asymptotic expansions of the inverse of the Beta distribution

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 ${\int }_0^q{t}^{a-1}(1-t{)}^{b-1}\text{d}t=pB(a,b)$, 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 Nørlund Polynomials, a generalisation of Bernoulli numbers. Finally, we provide Maple and Sage algorithms for computing the terms of the asymptotic expansions.

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