(Generalized) maximum cumulative direct, paired, and residual π entropy principle

Jaynes (1957a,b) formulates the maximum entropy (ME) principle as the search for a distribution maximizing a given entropy under some given constraints. Kapur (1984) and Kesavan & Kapur (1989) introduce the generalized maximum entropy principle as the derivation of an entropy for which a given distribution has the maximum entropy property under some given constraints. Both principles will be considered for cumulative entropies. Such entropies depend either on the distribution (direct) or on the survival function (residual) or on both (paired). Maximizing this entropy without any constraint gives a extremely U-shaped (= bipolar) distribution. Under the constraint of fixed mean and variance. maximizing the cumulative entropy 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 so-called contradictory information in contrast to minimum or no information. Only a few maximum entropy distributions for cumulative entropies have already been derived in the literature. We extend the results to well-known flexible distributions (like the generalized logistic distribution) and derive some special distributions (like the skewed logistic, the skewed Tukey Λ and the extended Burr XII distribution). The generalized maximum entropy principle will be applied to the generalized Tukey Λ distribution and the Fechner family of skewed distributions. At last, cumulative entropies will be estimated such that the data was drawn from a ME distribution. This estimator will be applied to the daily S&P500 returns and the time duration between mine explosions.