On Weighted Residual and Past Entropies

TL;DR

This paper introduces weighted residual and past entropies, which are length-biased, shift-dependent information measures that emphasize larger observed values, providing a dynamic perspective on the information of random lifetimes.

Contribution

It proposes new weighted entropy measures for random lifetimes, extending existing concepts with length-biased and shift-dependent properties, and analyzes their behavior under transformations.

Findings

Defined weighted residual and past entropies.

Analyzed their behavior under monotonic transformations.

Extended entropy concepts to dynamic lifetime analysis.

Abstract

We consider a "length-biased" shift-dependent information measure, related to the differential entropy in which higher weight is assigned to large values of observed random variables. This allows us to introduce the notions of "weighted residual entropy" and "weighted past entropy", that are suitable to describe dynamic information of random lifetimes, in analogy with the entropies of residual and past lifetimes introduced in [9] and [6], respectively. The obtained results include their behaviors under monotonic transformations.