Logarithmic concavity of the inverse incomplete beta function with respect to parameter

TL;DR

This paper analytically studies the inverse incomplete beta function as a function of its first parameter, establishing its logarithmic concavity and other properties, with implications for related distributions.

Contribution

It proves the logarithmic concavity of the inverse incomplete beta function and extends monotonicity results to a broader class of parametrized distributions.

Findings

Logarithmic concavity of the inverse incomplete beta function is established.

Monotonicity results are extended to inverses of a larger class of distributions.

Limit and convexity properties are characterized.

Abstract

The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.