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Maximum dynamic entropy models

Part of: Probability theory on algebraic and topological structures Distribution theory - Probability Communication, information

Published online by Cambridge University Press:  14 July 2016

Majid Asadi ,
Nader Ebrahimi ,
G. G. Hamedani  and
Ehsan S. Soofi
Majid Asadi*
Affiliation:
University of Isfahan
Nader Ebrahimi*
Affiliation:
Northern Illinois University
G. G. Hamedani*
Affiliation:
Marquette University
Ehsan S. Soofi*
Affiliation:
University of Wisconsin–Milwaukee
*
∗ Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran. Email address: [email protected]
∗∗ Postal address: Division of Statistics, Northern Illinois University, DeKalb, IL 60155, USA. Email address: [email protected]
∗∗∗ Postal address: Department of Mathematics, Statistics, and Computer Science, Marquette University, PO Box 1881, Milwaukee, WI 53201-1881, USA. Email address: [email protected]
∗∗∗∗ Postal address: School of Business Administration, University of Wisconsin–Milwaukee, PO Box 741, Milwaukee, WI 53201, USA. Email address: [email protected]
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Abstract

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.

Keywords

Lifetime distributionresidual life distributionmonotone densityfailure-rate dominanceuncertainty ordering

MSC classification

Primary: 94A15: Information theory, general
Secondary: 60E15: Inequalities; stochastic orderings 60B05: Probability measures on topological spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 379 - 390
Copyright
Copyright © Applied Probability Trust 2004 

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References

Di Crescenzo, A., and Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Prob. 39, 434–440.Google Scholar
Ebrahimi, N. (1996). How to measure uncertainty in the residual life distributions. Sankhyā A 58, 48–57.Google Scholar
Ebrahimi, N. (2000). The maximum entropy method for lifetime distributions. Sankhyā A 62, 223–235.Google Scholar
Ebrahimi, N., and Kirmani, S. N. U. A. (1997). Some results on ordering of survival functions through uncertainty. Statist. Prob. Lett. 11, 167–176.Google Scholar
Ebrahimi, N., and Pellery, F. (1995). New partial ordering of survival functions based on the notion of uncertainty. J. Appl. Prob. 23, 202–211.Google Scholar
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. 127–159.Google Scholar
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.Google Scholar
Jaynes, E. T. (1982). On the rationale of maximum entropy methods. Proc. IEEE 70, 939–952.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar
Shannon, C. E. (1948). A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423.CrossRefGoogle Scholar