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

TL;DR

This paper introduces 'pdf-related distributions' derived from random variables and their densities, analyzing their properties, stochastic comparisons, and implications for information measures like differential entropy and varentropy.

Contribution

It defines and studies the properties of pdf-related distributions, including their stochastic ordering and applications to information measures, providing new insights into their behavior.

Findings

Characterization of pdf-related distributions for exponential and Laplace types

Stochastic comparison methods for pdf-related distributions

Analysis of differential entropy and varentropy properties

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.