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Maximum dynamic entropy models

  • Majid Asadi (a1), Nader Ebrahimi (a2), G. G. Hamedani (a3) and Ehsan S. Soofi (a4)


A formal approach to produce a model for the data-generating distribution based on partial knowledge is the well-known maximum entropy method. In this approach, partial knowledge about the data-generating distribution is formulated in terms of some information constraints and the model is obtained by maximizing the Shannon entropy under these constraints. Frequently, in reliability analysis the problem of interest is the lifetime beyond an age t. In such cases, the distribution of interest for computing uncertainty and information is the residual distribution. The information functions involving a residual life distribution depend on t, and hence are dynamic. The maximum dynamic entropy (MDE) model is the distribution with the density that maximizes the dynamic entropy for all t. We provide a result that relates the orderings of dynamic entropy and the hazard function for distributions with monotone densities. Applications include dynamic entropy ordering within some parametric families of distributions, orderings of distributions of lifetimes of systems and their components connected in series and parallel, record values, and formulation of constraints for the MDE model in terms of the evolution paths of the hazard function and mean residual lifetime function. In particular, we identify classes of distributions in which some well-known distributions, including the mixture of two exponential distributions and the mixture of two Pareto distributions, are the MDE models.


Corresponding author

Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran. Email address:
∗∗ Postal address: Division of Statistics, Northern Illinois University, DeKalb, IL 60155, USA. Email address:
∗∗∗ Postal address: Department of Mathematics, Statistics, and Computer Science, Marquette University, PO Box 1881, Milwaukee, WI 53201-1881, USA. Email address:
∗∗∗∗ Postal address: School of Business Administration, University of Wisconsin–Milwaukee, PO Box 741, Milwaukee, WI 53201, USA. Email address:


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Di Crescenzo, A., and Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Prob. 39, 434440.
Ebrahimi, N. (1996). How to measure uncertainty in the residual life distributions. Sankhyā A 58, 4857.
Ebrahimi, N. (2000). The maximum entropy method for lifetime distributions. Sankhyā A 62, 223235.
Ebrahimi, N., and Kirmani, S. N. U. A. (1997). Some results on ordering of survival functions through uncertainty. Statist. Prob. Lett. 11, 167176.
Ebrahimi, N., and Pellery, F. (1995). New partial ordering of survival functions based on the notion of uncertainty. J. Appl. Prob. 23, 202211.
Ebrahimi, N., and Soofi, E. S. (2003). Information measures for reliability. In Mathematical Reliability: An Expository Perspective, eds Soyer, R., Mazzuchi, T. A. and Singpurwalla, N. D., Kluwer, Dordrecht, pp. 127159.
Ebrahimi, N., Hamedani, G. G., and Soofi, E. S. (1996). Maximum entropy modeling with partial information on failure rate. School of Business Administration, University of Wisconsin–Milwaukee.
Jaynes, E. T. (1982). On the rationale of maximum entropy methods. Proc. IEEE 70, 939952.
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, New York.
Shannon, C. E. (1948). A mathematical theory of communication. Bell Syst. Tech. J. 27, 379423.


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