(Generalized) Maximum Cumulative Direct, Residual, and Paired xcexa6 Entropy Approach A distribution that maximizes an entropy can be found by applying two different principles. On the one hand, Jaynes (1957a,b) formulated the maximum entropy principle (MaxEnt) as the search for a distribution maximizing a given entropy under some given constraints. On the other hand, Kapur (1994) and Kesavan and Kapur (1989) introduced the generalized maximum entropy principle (GMaxEnt) as the derivation of an entropy for which a given distribution has the maximum entropy property under some given constraints. In this paper, both principles were considered for cumulative entropies. Such entropies depend either on the distribution function (direct), on the survival function (residual) or on both (paired). We incorporate cumulative direct, residual, and paired entropies in one approach called cumulative    xcexa6    entropies. Maximizing this entropy without any constraints produces an extremely Ushaped (bipolar) distribution. Maximizing the cumulative entropy under the constraints of fixed mean and variance tries to transform a distribution in the direction of a bipolar distribution, as far as it is allowed by the constraints. A bipolar distribution represents socalled contradictory information, which is in contrast to minimum or no information. In the literature, to date, only a few maximum entropy distributions for cumulative entropies have been derived. In this paper, we extended the results to well known flexible distributions (like the generalized logistic distribution) and derived some special distributions (like the skewed logistic, the skewed Tukey    xcexbb    and the extended Burr XII distribution). The generalized maximum entropy principle was applied to the generalized Tukey    xcexbb    distribution and the Fechner family of skewed distributions. Finally, cumulative entropies were estimated such that the data was drawn from a maximum entropy distribution. This estimator will be applied to the daily Su0026P500 returns and time durations between mine explosions.

On properties of a new decomposable entropy of DempsterShafer belief functions Abstract   We define entropy of belief functions in the DempsterShafer (DS) theory that satisfies a compound distributions property that is analogous to the property that characterizes Shannonu0027s definitions of entropy and conditional entropy for probability mass functions. None of the existing definitions of entropy for belief functions in the DS theory satisfy this property. We describe some important properties of our definition, and discuss its semantics as a measure of dissonance and not uncertainty. Finally, we compare our definition of entropy with some other definitions that are similar to ours in the sense that these definitions measure dissonance and not uncertainty.