The Bunching and Monotonicity Properties of Families of Probability Distributions

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.