Extension of the past lifetime and its connection to the cumulative entropy

TL;DR

This paper explores the reversed relevation transform for lifetime distributions, establishing its properties, connections to cumulative entropy, and applications to system lifetimes and stochastic ordering.

Contribution

It introduces the reversed relevation transform, analyzes its properties, and links it to cumulative entropy and stochastic orders, providing new tools for lifetime analysis.

Findings

Reversal transform is commutative under proportional hazard models.

Constructs a sequence of random variables linked to cumulative entropy.

Provides iterative formulas for mean and entropy calculations.

Abstract

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazards rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.