# Divergence from, and Convergence to, Uniformity of Probability Density Quantiles

## TL;DR

This paper studies how probability density quantiles evolve under repeated transformations, establishing convergence to uniformity and linking divergence measures to statistical tests for shape and tail behavior.

## Contribution

It introduces new fixed point theorems for pdQ convergence and connects divergence measures to optimal uniformity tests.

## Key findings

- Repeated pdQ applications converge to uniform distribution.
- Kullback-Leibler divergence informs power of uniformity tests.
- New fixed point theorems for pdQ mappings.

## Abstract

The probability density quantile (pdQ) carries essential information regarding shape and tail behavior of a location-scale family. Convergence of repeated applications of the pdQ mapping to the uniform distribution is investigated and new fixed point theorems are established. The Kullback-Leibler divergences from uniformity of these pdQs are mapped and found to be ingredients in power functions of optimal tests for uniformity against alternative shapes.