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# (Generalized) Maximum Cumulative Direct, Residual, and Paired Φ Entropy Approach

by Ingo Klein 1,* and Monika Doll 2
1
Department of Statistics and Econometrics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Lange Gasse 20, 90403 Nürnberg, Germany
2
GfK SE, Nordwestring 101, 90419 Nürnberg, Germany
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(1), 91; https://doi.org/10.3390/e22010091
Received: 18 December 2019 / Revised: 4 January 2020 / Accepted: 9 January 2020 / Published: 12 January 2020
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 Φ entropies. Maximizing this entropy without any constraints produces an extremely U-shaped (=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 so-called 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 $λ$ and the extended Burr XII distribution). The generalized maximum entropy principle was applied to the generalized Tukey $λ$ 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 S&P500 returns and time durations between mine explosions.
MDPI and ACS Style

Klein, I.; Doll, M. (Generalized) Maximum Cumulative Direct, Residual, and Paired Φ Entropy Approach. Entropy 2020, 22, 91.