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An easily verifiable dispersion order for discrete distributions

Andreas Eberl, Bernhard Klar, Alfonso Suárez-Llorens

TL;DR

This work addresses the limitations of the classical dispersive order for discrete distributions by introducing a weaker, practically applicable weak dispersive order based on the Lévy concentration function $Q_X(\varepsilon)$. It formalizes both the order and a family of concentration-based variability measures $\nu_r(X)$ and $\nu_{\text{rob}}(X)$ that satisfy the Bickel–Lehmann axioms and respect the weak dispersive relation, linking to majorization and entropy concepts. The paper proves that $F \leq_{disp}^{\land\text{-disc}} G$ implies $F \leq_{wd} G$, yet the weak order is strictly weaker, and provides empirical illustrations showing when wd-based conclusions align or diverge from traditional dispersion measures. Overall, the approach broadens the toolkit for comparing dispersion in discrete data and suggests robust, interpretable alternatives for applied statistics, with potential extensions to multivariate settings and deeper theoretical connections between concentration and variability.

Abstract

Dispersion is a fundamental concept in statistics, yet standard approaches - especially via stochastic orders - face limitations in the discrete setting. In particular, the classical dispersive order, well-established for continuous distributions, becomes overly restrictive for discrete random variables due to support inclusion requirements. To address this, we propose a novel weak dispersive order for discrete distributions. This order retains desirable properties while relaxing structural constraints, thereby broadening applicability. We further introduce a class of variability measures based on probability concentration, offering robust and interpretable alternatives that conform to classical axioms. Empirical illustrations highlight the practical relevance of this framework.

An easily verifiable dispersion order for discrete distributions

TL;DR

This work addresses the limitations of the classical dispersive order for discrete distributions by introducing a weaker, practically applicable weak dispersive order based on the Lévy concentration function . It formalizes both the order and a family of concentration-based variability measures and that satisfy the Bickel–Lehmann axioms and respect the weak dispersive relation, linking to majorization and entropy concepts. The paper proves that implies , yet the weak order is strictly weaker, and provides empirical illustrations showing when wd-based conclusions align or diverge from traditional dispersion measures. Overall, the approach broadens the toolkit for comparing dispersion in discrete data and suggests robust, interpretable alternatives for applied statistics, with potential extensions to multivariate settings and deeper theoretical connections between concentration and variability.

Abstract

Dispersion is a fundamental concept in statistics, yet standard approaches - especially via stochastic orders - face limitations in the discrete setting. In particular, the classical dispersive order, well-established for continuous distributions, becomes overly restrictive for discrete random variables due to support inclusion requirements. To address this, we propose a novel weak dispersive order for discrete distributions. This order retains desirable properties while relaxing structural constraints, thereby broadening applicability. We further introduce a class of variability measures based on probability concentration, offering robust and interpretable alternatives that conform to classical axioms. Empirical illustrations highlight the practical relevance of this framework.

Paper Structure

This paper contains 11 sections, 8 theorems, 30 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. The weak dispersive order for discrete distributions
  3. Properties of the weak dispersive order
  4. Relation to the discrete dispersive order of Eberl and Klar
  5. New measures of variability based on the concentration function
  6. Empirical illustrations
  7. Counts of swimbladder nematodes in Japanese eels
  8. Counts of swimbladder nematodes in European eels
  9. Counts of intestinal parasites in European eels
  10. Amount of aggression attributed to film characters
  11. Concluding remarks

Key Result

Theorem 6

Let $X$ and $Y$ be two random variables such that $X=_{st} \phi(Y)$, where $\phi$ is a monotone contraction function. Then $X\leq_{\text{wd}}Y$.

Figures (4)

  • Figure 2: Bar plots of the relative frequencies for the datasets in Example \ref{['ex1']}
  • Figure 3: Bar plots of the relative frequencies for the datasets in Example \ref{['ex2']}
  • Figure 4: Bar plots of the relative frequencies for the datasets in Example \ref{['ex3']}
  • Figure 5: Bar plots of the relative frequencies for the datasets in Example \ref{['ex4']}

Theorems & Definitions (33)

  • Remark 1
  • Definition 2
  • Remark 3
  • proof
  • Definition 5
  • Theorem 6
  • proof
  • Remark 7
  • Corollary 8
  • proof
  • ...and 23 more