The Shapes of Things to Come: Probability Density Quantiles

TL;DR

This paper introduces probability density quantiles (pdQs) as a unique, shape-focused way to compare and classify probability distributions, providing robust estimation and tail behavior analysis.

Contribution

It develops a novel shape comparison method using pdQs, including estimation procedures and shape classification based on tail behavior.

Findings

Empirical estimates of pdQs are consistent and robust.

Shape differences can be quantified using Hellinger or Kullback-Leibler measures.

Tail behavior classification is simplified through boundary derivatives of pdQs.

Abstract

For every discrete or continuous location-scale family having a square-integrable density, there is a unique continuous probability distribution on the unit interval that is determined by the density-quantile composition introduced by Parzen in 1979. These probability density quantiles (pdQs) only differ in shape, and can be usefully compared with the Hellinger distance or Kullback-Leibler divergences. Convergent empirical estimates of these pdQs are provided, which leads to a robust global fitting procedure of shape families to data. Asymmetry can be measured in terms of distance or divergence of pdQs from the symmetric class. Further, a precise classification of shapes by tail behavior can be defined simply in terms of pdQ boundary derivatives.