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

Part of: Survival analysis and censored data Distribution theory - Probability Sufficiency and information Communication, information

Published online by Cambridge University Press:  30 March 2016

Antonio Di Crescenzo  and
Abdolsaeed Toomaj
Antonio Di Crescenzo*
Affiliation:
Università degli Studi di Salerno
Abdolsaeed Toomaj*
Affiliation:
Gonbad Kavous University
*
∗Postal address: Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, I-84084 Fisciano (SA), Italy. Email address: adicrescenzo@unisa.it
∗∗Postal address: Department of Statistics, Gonbad Kavous University, Gonbad Kavous, Iran. Email address: ab.toomaj@gmail.com
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Abstract

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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.

Keywords

Relevation transformreversed relevation transformproportional hazard rate modelproportional reversed hazard rate modelweighted cumulative distributioncumulative entropy

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62B10: Information-theoretic topics 62N05: Reliability and life testing 94A17: Measures of information, entropy
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1156 - 1174
Copyright
Copyright © Applied Probability Trust 2015 

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