Why Most Empirical Distributions Are Few-Modal

Authors

Julio Urenda, The University of Texas at El PasoFollow
Olga Kosheleva, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

5-2020

Comments

Technical Report: UTEP-CS-20-50

Abstract

In principle, any non-negative function can serve as a probability density function -- provided that it adds up to 1. All kinds of processes are possible, so it seems reasonable to expect that observed probability density functions are random with respect to some appropriate probability measure on the set of all such functions -- and for all such measures, similarly to the simplest case of random walk, almost all functions have infinitely many local maxima and minima. However, in practice, most empirical distributions have only a few local maxima and minima -- often one (unimodal distribution), sometimes two (bimodal), and, in general, they are few-modal. From this viewpoint, econometrics is no exception: empirical distributions of economics-related quantities are also usually few-modal. In this paper, we provide a theoretical explanation for this empirical fact.