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

Abstract

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.

Generalized Entropies, Variance and Applications

“…See Klein et al [ 45 ] and Klein and Doll [ 46 ] for detailed investigations on , also dealing with relationships of the cumulative paired entropy to the variance. This entropy has the properties of a measure of scale (or dispersion).…”

Measuring Dispersion and Serial Dependence in Ordinal Time Series Based on the Cumulative Paired ϕ-Entropy

“…Building upon earlier works by Klein [ 16 ], Yager [ 17 ], it was recently shown by Klein Doll [ 18 ], Klein et al [ 19 ] that the aforementioned ordinal dispersion measures can be subsumed under the family of so-called “cumulative paired -entropies” (see Section 2 ), abbreviated as , which constitutes the starting point of the present article. Our first main task is to derive the asymptotic distribution of the corresponding sample version, , for both i. i. d. and time series data, and to check the finite sample performance of the resulting approximate distribution, see Section 3 and Section 5 .…”

“…Klein Doll [ 18 ], Klein et al [ 19 ] proposed and investigated a family of cumulative paired -entropies. Although their main focus was on continuously distributed random variables, they also referred to the ordinal case and pointed out that many well-known ordinal dispersion measures are included in this family.…”

On reconsidering entropies and divergences and their cumulative counterparts: Csiszár's, DPD's and Fisher's type cumulative and survival measures

“…These papers were the basis for an increasing interest for the definition of informational measures in the context of the paper by Rao et al [73]. In this direction, entropy and divergence type measures are introduced and studied, in the framework of the cumulative distribution function or in terms of the respective survival function, in the papers by Rao [72], Zografos and Nadarajah [98], Di Crescenzo and Longobardi [31], Baratpour and Rad [12], Park et al [66] and the subsequent papers by Di Crescenzo and Longobardi [32], Klein et al [46], Asadi et al [6,7], Park et al [67], Klein and Doll [45], among many others. In these and other treatments, entropy type measures and Kullback-Leibler type divergences have been mainly received the attention of the authors.…”

“…In these and other treatments, entropy type measures and Kullback-Leibler type divergences have been mainly received the attention of the authors. Entropy and divergence type measures are also considered in the papers by Klein et al [46], Asadi et al [6], Klein and Doll [45] by combining the cumulative distribution function and the survival function. However, to the best of our knowledge, it seems that there has not yet appeared in the existing literature a definition of the broad class of Csiszár's type -divergences or a definition of the density power divergence in the framework which was initiated in the paper by Rao et al [73].…”