Fractional generalized cumulative entropy and its dynamic version

TL;DR

This paper introduces fractional generalized cumulative entropy and its dynamic version, extending information measures to better analyze distributions, especially in reliability contexts, with theoretical properties and empirical estimation methods.

Contribution

It proposes a new fractional entropy measure based on the cumulative distribution function, along with its dynamic version and non-parametric estimator, connecting to reliability theory and fractional calculus.

Findings

The new measure is a variability measure suitable for proportional reversed hazard models.

The empirical estimator converges almost surely to the theoretical measure.

A central limit theorem is established under exponential distribution.

Abstract

Following the theory of information measures based on the cumulative distribution function, we propose the fractional generalized cumulative entropy, and its dynamic version. These entropies are particularly suitable to deal with distributions satisfying the proportional reversed hazard model. We study the connection with fractional integrals, and some bounds and comparisons based on stochastic orderings, that allow to show that the proposed measure is actually a variability measure. The investigation also involves various notions of reliability theory, since the considered dynamic measure is a suitable extension of the mean inactivity time. We also introduce the empirical generalized fractional cumulative entropy as a non-parametric estimator of the new measure. It is shown that the empirical measure converges to the proposed notion almost surely. Then, we address the stability of the empirical measure and provide an example of application to real data. Finally, a central limit theorem is established under the exponential distribution.