Quantile characterization of univariate unimodality

TL;DR

The paper develops a quantile-based characterization of univariate unimodality by linking the quantile function $Q_\mu$ and its density $q_\mu$ to the classical CDF/PMF view. It proves that unimodality is equivalent to the quasi-convexity of the quantile density and to the concave-convex structure of $Q_\mu$ around a quantile mode $\alpha$, with $Q_\mu$ being absolutely continuous and related to the density via $f_{abs}=\frac{1}{q_\mu\circ F_\mu}$ a.e. on the support. The work relies on a general theory of non-decreasing functions, generalized inverses, and RN derivatives, providing an inverse function rule $g_{abs}=\frac{1}{h\circ G}$ a.e. for absolutely continuous inverses and extending unimodality results from CDFs to locally finite measures. These insights connect density shapes, CDF geometry, and quantile-based representations, offering a simple, robust framework with implications for Wasserstein-2 barycenters and the preservation of unimodality under operations on distributions.

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.

Figures (2)

• Figure 1: The distribution $\mu$ is CDF-unimodal: The density $f_{abs}$ (left, dashed) of $\mu_{abs}$, the absolute continuous part of $\mu$, is quasi-concave (non-decreasing on $A=(-\infty,\frac{1}{2})$ and non-increasing on $B=[\frac{1}{2},+\infty)$), hence, $\mu_{abs}$ is dens-unimodal. The CDF $F_\mu$ (left, filled) is convex on $(-\infty,\frac{1}{2}]$ and concave on $[\frac{1}{2},+\infty)$. The quantile density $q_\mu$ (right, dashed) is quasi-convex with modal interval $[\alpha_{\min},\alpha_{\max}]=[\frac{1}{3},\frac{2}{3}]$ (non-increasing on $(0,\alpha)$ and non-decreasing on $[\alpha,1)$ for all $\alpha\in[\alpha_{\min},\alpha_{\max}]$). The QF $Q_\mu$ (right, filled) is concave on $(0,\alpha]$ and convex on $[\alpha,1)$, for $\alpha\in[\alpha_{\min},\alpha_{\max}]$. Empty circles denote left or right limits, while the filled circles show the values attained by the respective functions at their discontinuities. The choices for the densities were made arbitrary.
• Figure 2: The right-continuous version $G_r$ (orange, filled, transparent) of a non-decreasing function with its real-valued restriction $\Tilde{G}_r:I_G\to\mathbb{R}$ (orange, filled) are plotted as functions mapping $x$ values to $y$ values while a generalized inverse $H=\frac{H_l+H_r}{2}$ (blue, dashed, transparent) of $G$ and its real restriction $\Tilde{H}:I_H\to\mathbb{R}$ (blue, dashed) are plotted as functions mapping $y$ values to $x$ values. The left and right limits of $G_r$, $H$ are shown as empty circles, respectively empty squares. The filled variants represent the value attained by the function at a discontinuity. The braces visualize the intervals: $S_G=\overline{{G_r}^{-1}\parenthesis[-NoValue-]{M_H}}$, $M_H=\parenthesis[-NoValue-]{{H}^{-1}\parenthesis[-NoValue-]{I_G}}^\circ$, $I_G=\parenthesis[-NoValue-]{{G_r}^{-1}\parenthesis[-NoValue-]{\mathbb{R}}}^\circ$. We have that $\mu_G=\delta_0+ \operatorname{Beta} \IfNoValueTF{-NoValue-} {\IfNoValueF{2,2} { \parenthesis[-NoValue-] {2,2} } } { \parenthesis[-NoValue-]{-NoValue-} \IfNoValueF{2,2}{(2,2)} }$ restricted to $(-\infty,2)=I_G$.

Theorems & Definitions (35)

• Definition 1.1
• Definition 1.2
• Definition 1.3: Dharmadhikari.1988
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4