Maximum dynamic entropy models

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.

Some Results on Dynamic Discrimination Measures of Order (alpha, beta)

“…Several researchers have studied KL discrimination measure by including the current age. In this direction we refer to Asadi et al [2], Di Crescenzo and Longobardi [7] and Ebrahimi and Kirmani [10,11]. Later the discrimination measure (1.1) was generalized, called discrimination measure of order α, as As similar measure to (1.2), discrimination measure of order α between two rv's X and Y at time t can be defined by (see Asadi et al [3]) …”

A Note on the Bivariate Maximum Entropy Modeling

“…The present paper provides a solution for problem of specification of bivariate models using the well-known Principle of Maximum Entropy; especially when partial information about the dependence structures between X 1 and X 2 are available and the constraints are made based on hazard gradient or reversed hazard gradient. In the univariate case, when the constraints are based on hazard rate and mean residual life function, Asadi et al (2004) studied a concept of maximum dynamic entropy and Asadi et al (2005) introduced a notion of minimum dynamic discrimination information and obtained various univariate lifetime distributions as maximum dynamic entropy and minimum dynamic discrimination models. Recently, Asadi et al (2010) have studied a concept of bivariate dynamic ME model and derived several bivariate distributions when partial information is available on hazard gradient.…”

Minimum dynamic discrimination information models

“…Recently, a maximum dynamic entropy (MDE) procedure for developing lifetime models has been proposed. This may be viewed as an extension of the ME principle in the case that the information is given in terms of hazard rate growth inequality constraints (Asadi et al (2004)). The ME and MDI procedures (i.e.…”

“…Several authors have studied information functions that take age into account (Ebrahimi (1996), Ebrahimi and Kirmani (1996a), (1996b), Di Crescenzo and Longobardi (2002, Minimum dynamic discrimination information models 645 Belzunce et al (2004), Asadi et al (2004)). Consideration of age has led to some important insights about lifetime models, such as an information characterization of the proportional hazards model (Ebrahimi and Kirmani (1996a)) and MDE characterizations of various lifetime models, including some mixture distributions, for which no other maximum entropy formulation is available (Asadi et al (2004)).…”

“…Consideration of age has led to some important insights about lifetime models, such as an information characterization of the proportional hazards model (Ebrahimi and Kirmani (1996a)) and MDE characterizations of various lifetime models, including some mixture distributions, for which no other maximum entropy formulation is available (Asadi et al (2004)). Some tests of distributional hypotheses based on dynamic information measures have been developed for reliability analysis (Ebrahimi (1998), (2001)).…”