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On dynamic mutual information for bivariate lifetimes

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

Published online by Cambridge University Press:  21 March 2016

Jafar Ahmadi ,
Antonio Di Crescenzo  and
Maria Longobardi
Jafar Ahmadi*
Affiliation:
Ferdowsi University of Mashhad
Antonio Di Crescenzo*
Affiliation:
Università degli Studi di Salerno
Maria Longobardi*
Affiliation:
Università di Napoli Federico II
*
∗ Postal address: Department of Statistics, Ordered and Spatial Data Center of Excellence, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad, 91775, Iran. Email address: [email protected]
∗∗ Postal address: Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy. Email address: [email protected]
∗∗∗ Postal address: Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, 80126 Napoli, Italy. Email address: [email protected]
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Abstract

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We consider dynamic versions of the mutual information of lifetime distributions, with a focus on past lifetimes, residual lifetimes, and mixed lifetimes evaluated at different instants. This allows us to study multicomponent systems, by measuring the dependence in conditional lifetimes of two components having possibly different ages. We provide some bounds, and investigate the mutual information of residual lifetimes within the time-transformed exponential model (under both the assumptions of unbounded and truncated lifetimes). Moreover, with reference to the order statistics of a random sample, we evaluate explicitly the mutual information between the minimum and the maximum, conditional on inspection at different times, and show that it is distribution-free in a special case. Finally, we develop a copula-based approach aiming to express the dynamic mutual information for past and residual bivariate lifetimes in an alternative way.

Keywords

Entropymutual informationtime-transformed exponential modelbivariate lifetimesorder statisticscopula

MSC classification

Primary: 94A17: Measures of information, entropy
Secondary: 62N05: Reliability and life testing 60E99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 1157 - 1174
Copyright
Copyright © Applied Probability Trust 2015 

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