Stochastic comparisons, differential entropy and varentropy for distributions induced by probability density functions

TL;DR

This work introduces pdf-related distributions, defined by transforming an absolutely continuous random variable through its own density, to study distribution forms, quantiles, and their relation to reliability and information measures. By leveraging unimodality and a suite of stochastic orders (usual, dispersive, convex transform, star, kurtosis) as well as density rearrangements, the authors develop a framework to compare pdf-related distributions and derive implications for differential entropy $H(X)$ and varentropy $V(X)$, including residual lifetimes $X_t$. They provide characterizations for when $f(X)$ is uniform on $(0,1)$ and demonstrate how orderings propagate to $H$ and $V$, with applications to Weibull and Pareto-type models. The results yield practical tools for evaluating and bounding information measures across distributions, with potential impact on reliability analysis and information-theoretic comparisons of continuous variables.

Abstract

Stimulated by the need of describing useful notions related to information measures, we introduce the `pdf-related distributions'. These are defined in terms of transformation of absolutely continuous random variables through their own probability density functions. We investigate their main characteristics, with reference to the general form of the distribution, the quantiles, and some related notions of reliability theory. This allows us to obtain a characterization of the pdf-related distribution being uniform for distributions of exponential and Laplace type as well. We also face the problem of stochastic comparing the pdf-related distributions by resorting to suitable stochastic orders. Finally, the given results are used to analyse properties and to compare some useful information measures, such as the differential entropy and the varentropy.