Journal of Applied Probability

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

Antonio Di Crescenzo and Abdolsaeed Toomaj

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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

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

First available in Project Euclid: 22 December 2015

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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

Relevation transform reversed relevation transform proportional hazard rate model proportional reversed hazard rate model weighted cumulative distribution cumulative 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.

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