Defining Dispersion: A Fundamental Order for Univariate Discrete Distributions

TL;DR

This paper introduces discrete adaptations of the dispersive order to better measure dispersion in discrete distributions, addressing limitations of traditional measures and ensuring consistency with continuous cases.

Contribution

It defines and analyzes discrete dispersive orders, aligning them with continuous properties and evaluating popular dispersion measures' effectiveness in discrete contexts.

Findings

Discrete dispersive orders are validated as a foundation for discrete dispersion measures.

Most popular dispersion measures preserve the discrete dispersive orders.

Interquantile range does not preserve discrete dispersive orders, making it unsuitable.

Abstract

The measurement of dispersion is one of the most fundamental and ubiquitous statistical concepts, in both applied and theoretical contexts. For dispersion measures, such as the standard deviation, to effectively capture the variability of a given distribution, they must, by definition, preserve some stochastic order of dispersion. The so-called dispersive order is the most basic order that serves as a foundation underneath the concept of dispersion measures. However, this order is incompatible with almost all discrete distributions, including lattice and most empirical distributions. As a result, popular measures may fail to accurately capture the dispersion of such distributions. In this paper, discrete adaptations of the dispersive order are defined and analyzed. They are shown to be a compromise between being equivalent to the original dispersive order on their joint area of applicability and other crucial properties. Moreover, they share many characteristic properties with the dispersive order, validating their role as a foundation for measuring discrete dispersion in a manner closely aligned with the continuous setting. Their behaviour on well-known families of lattice distribution is generally as expected when parameter differences are sufficiently large. Most popular dispersion measures preserve both discrete dispersive orders, rigorously ensuring that they are also meaningful in discrete settings. However, the interquantile range fails to preserve either discrete order, indicating that it is unsuitable for measuring the dispersion of discrete distributions.