Convex transform order of Beta distributions with some consequences

TL;DR

This paper characterizes when one Beta distribution is smaller than another in the convex transform order, leading to new inequalities and monotonicity results related to skewness, modes, and distribution function behaviors.

Contribution

It provides a complete characterization of the convex transform order for Beta distributions and derives new inequalities and properties related to skewness and distribution functions.

Findings

Complete characterization of convex transform order for Beta distributions

Monotonicity properties of probabilities exceeding mean or mode

New inequalities for skewed unimodal distributions and Beta cases

Abstract

The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterisation of when one Beta distribution is smaller than another according to the convex transform order. As an application, we derive monotonicity properties for the probability of Beta distributed random variables exceeding the mean or mode of their distribution. Moreover, we obtain a simple alternative proof of the mode-median-mean inequality for unimodal distributions that are skewed in a sense made precise by the convex transform order. This new proof also gives an analogous inequality for the anti-mode of distributions that have a unique anti-mode. Such inequalities for Beta distributions follow as special cases. Finally, some consequences for the values of distribution functions of Binomial distributions near to their means are mentioned.