Given two absolutely continuous nonnegative independent random variables, wedefine the reversed relevation transform as dual to the relevation transform.We first apply such transforms to the lifetimes of the components of paralleland series systems under suitably proportionality assumptions on the hazardrates. Furthermore, we prove that the (reversed) relevation transform iscommutative if and only if the proportional (reversed) hazard rate model holds.By repeated application of the reversed relevation transform we construct adecreasing sequence of random variables which leads to new weighted probabilitydensities. We obtain various relations involving ageing notions and stochasticorders. We also exploit the connection of such a sequence to the cumulativeentropy and to an operator that is dual to the Dickson-Hipp operator. Iterativeformulae for computing the mean and the cumulative entropy of the randomvariables of the sequence are finally investigated.
"Extension of the past lifetime and its connection to the cumulative entropy." J. Appl. Probab. 52 (4) 1156 - 1174, December 2015. https://doi.org/10.1239/jap/1450802759