Journal of Applied Probability
- J. Appl. Probab.
- Volume 52, Number 4 (2015), 1156-1174.
Extension of the past lifetime and its connection to the cumulative entropy
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.
J. Appl. Probab., Volume 52, Number 4 (2015), 1156-1174.
First available in Project Euclid: 22 December 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62B10: Information-theoretic topics [See also 94A17] 62N05: Reliability and life testing [See also 90B25] 94A17: Measures of information, entropy
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