Asymptotic expansions of the inverse of the Beta distribution

2016/11/11 by Dimitris Askitis, Askitis, Dimitris
Mathematics · #11B68 (Secondary ) #33B15 #41A60 (Primary) #60E05 #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1611.03573

openalex publication_date 2016/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∫₀^q t^{a - 1} (1 - t)^{b - 1} 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 ∞. 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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