The Bunching and Monotonicity Properties of Families of Probability Distributions

TL;DR

This paper introduces a new way to measure concentration and stochastic order in probability distributions, applies it to Beta distributions, and uses income data to assess income concentration.

Contribution

It develops a novel concentration measure and stochastic order concept, with applications to Beta distributions and income data analysis.

Findings

New concentration measure for continuous variables

Application to Beta distributions with constant mean

U.S. income data fitted with generalized Beta distributions

Abstract

Measuring the concentration of random variables is a fundamental concept in probability and statistics. Here, we explore a type of concentration measure for continuous random variables with bounded support and use it to provide a notion of stochastic order by concentration. We give an application to the Beta family of distributions, and specifically to the one-parameter subfamily with constant mean. This leads to using U.S. household income data to fit generalized Beta distributions and offers a new measure of income concentration.