Quantile characterization of univariate unimodality

TL;DR

This paper characterizes univariate unimodal distributions through properties of their quantile functions, showing that unimodality corresponds to the quasi-convexity of the quantile density, with implications for distribution analysis.

Contribution

It introduces a novel shape constraint characterization of unimodal distributions via their quantile functions, linking unimodality to the quasi-convexity of the quantile density.

Findings

Quantile functions of unimodal distributions are always absolutely continuous.

Unimodality is equivalent to the quasi-convexity of the quantile density.

The analysis uses generalized inverse functions and inverse function rules.

Abstract

Unimodal univariate distributions can be characterized as piecewise convex-concave cumulative distribution functions. In this note we transfer this shape constraint characterization to the quantile function. We show that this characterization comes with the upside that the quantile function of a unimodal distribution is always absolutely continuous and consequently unimodality is equivalent to the quasi-convexity of its Radon-Nikodym derivative, i.e., the quantile density. Our analysis is based on the theory of generalized inverses of non-decreasing functions and relies on a version of the inverse function rule for non-decreasing functions.