## Journal of Applied Probability

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

#### 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 hazard 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.

#### Article information

Source
J. Appl. Probab., Volume 52, Number 4 (2015), 1156-1174.

Dates
First available in Project Euclid: 22 December 2015

https://projecteuclid.org/euclid.jap/1450802759

Digital Object Identifier
doi:10.1239/jap/1450802759

Mathematical Reviews number (MathSciNet)
MR3439178

Zentralblatt MATH identifier
1336.60029

#### Citation

Di Crescenzo, Antonio; Toomaj, Abdolsaeed. Extension of the past lifetime and its connection to the cumulative entropy. J. Appl. Probab. 52 (2015), no. 4, 1156--1174. doi:10.1239/jap/1450802759. https://projecteuclid.org/euclid.jap/1450802759

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