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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 

References

Abbasnejad, M., Arghami, N. R., Morgenthaler, S. and Borzadaran, G. R. Mohtashami (2010). On the dynamic survival entropy. Statist. Prob. Lett. 80, 1962–1971.CrossRefGoogle Scholar
Arellano-Valle, R. B., Contreras-Reyes, J. E. and Genton, M. G. (2013). Shannon entropy and mutual information for multivariate skew-elliptical distributions. Scand. J. Statist. 40, 42–62.CrossRefGoogle Scholar
Asadi, M. and Zohrevand, Y. (2007). On the dynamic cumulative residual entropy. J. Statist. Planning Infer. 137, 1931–1941.CrossRefGoogle Scholar
Bassan, B. and Spizzichino, F. (2005). Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93, 313–339.CrossRefGoogle Scholar
Belzunce, F., Navarro, J., Ruiz, J. M. and del Aguila, Y. (2004). Some results on residual entropy function. Metrika 59, 147–161.CrossRefGoogle Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Davy, M. and Doucet, A. (2003). Copulas: a new insight into positive time-frequency distributions. IEEE Signal Process. Lett. 10, 215–218.CrossRefGoogle Scholar
Di Crescenzo, A. and Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Prob. 39, 434–440.CrossRefGoogle Scholar
Di Crescenzo, A. and Longobardi, M. (2009). On cumulative entropies. J. Statist. Planning. Infer. 139, 4072–4087.CrossRefGoogle Scholar
Di Crescenzo, A., Longobardi, M. and Nastro, A. (2004). On the mutual information between residual lifetime distributions. In: Cybernetics and Systems 2004, Vol. 1, Austrian Society for Cybernetic Studies, Vienna, pp. 142–145.Google Scholar
Durante, F. and Sempi, C. (2015). Principles of Copula Theory. Chapman and Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Ebrahimi, N. (1996). How to measure uncertainty in the residual life time distribution. Sankhyā A 58, 48–56.Google Scholar
Ebrahimi, N. and Pellerey, F. (1995). New partial ordering of survival functions based on the notion of uncertainty. J. Appl. Prob. 32, 202–211.CrossRefGoogle Scholar
Ebrahimi, N., Kirmani, S. N. U. A. and Soofi, E. S. (2007). Multivariate dynamic information. J. Multivariate Anal. 98, 328–349.CrossRefGoogle Scholar
Ebrahimi, N., Soofi, E. S. and Soyer, R. (2008). Multivariate maximum entropy identification, transformation, and dependence. J. Multivariate Anal. 99, 1217–1231.CrossRefGoogle Scholar
Ebrahimi, N., Soofi, E. S. and Soyer, R. (2010). Information measures in perspective. Internat. Statist. Rev. 78, 383–412.CrossRefGoogle Scholar
Ebrahimi, N., Soofi, E. S. and Zahedi, H. (2004). Information properties of order statistics and spacings. IEEE Trans. Inf. Theory 50, 177–183.CrossRefGoogle Scholar
Ebrahimi, N., Hamedani, G. G., Soofi, E. S. and Volkmer, H. (2010). A class of models for uncorrelated random variables. J. Multivariate Anal. 101, 1859–1871.CrossRefGoogle Scholar
Giraudo, M. T., Sacerdote, L. and Sirovich, R. (2013). Non-parametric estimation of mutual information through the entropy of the linkage. Entropy 15, 5154–5177.CrossRefGoogle Scholar
Gupta, R. C. (2008). Reliability studies of bivariate distributions with exponential conditionals. Math. Comput. Modelling 47, 1009–1018.CrossRefGoogle Scholar
Kundu, C., Nanda, A. K. and Maiti, S. S. (2010). Some distributional results through past entropy. J. Statist. Planning Infer. 140, 1280–1291.CrossRefGoogle Scholar
Li, X. and Lin, J. (2011). Stochastic orders in time transformed exponential models with applications. Insurance Math. Econom. 49, 47–52.CrossRefGoogle Scholar
Misagh, F. and Yari, G. H. (2011). On weighted interval entropy. Statist. Prob. Lett. 81, 188–194.CrossRefGoogle Scholar
Mulero, J., Pellerey, F. and Rodríguez-Griñolo, R. (2010). Stochastic comparisons for time transformed exponential models. Insurance Math. Econom. 46, 328–333.CrossRefGoogle Scholar
Nanda, A. K. and Paul, P. (2006). Some properties of past entropy and their applications. Metrika 64, 47–61.CrossRefGoogle Scholar
Nanda, A. K. and Paul, P. (2006). Some results on generalized past entropy. J. Statist. Planning Infer. 136, 3659–3674.CrossRefGoogle Scholar
Navarro, J., del Aguila, Y. and Asadi, M. (2010). Some new results on the cumulative residual entropy.break J. Statist. Planning Infer. 140, 310–322.CrossRefGoogle Scholar
Pellerey, F. (2008). On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas. Kybernetika 44, 795–806.Google Scholar
Roy, D. (2002). On bivariate lack of memory property and a new definition. Ann. Inst. Statist. Math. 54, 404–410.CrossRefGoogle Scholar
Sunoj, S. M. and Linu, M. N. (2012). Dynamic cumulative residual Renyi's entropy. Statistics 46, 41–56.CrossRefGoogle Scholar
You, Y., Li, X. and Balakrishnan, N. (2014). On extremes of bivariate residual lifetimes from generalized Marshall–Olkin and time transformed exponential models. Metrika 77, 1041–1056.CrossRefGoogle Scholar