On weighted cumulative residual entropy

Covariance Representations and Coherent Measures for Some Entropies

We obtain covariance and Choquet integral representations for some entropies and give upper bounds of those entropies. The coherent properties of those entropies are discussed. Furthermore, we propose tail-based cumulative residual Tsallis entropy of order α (TCRTE) and tail-based right-tail deviation (TRTD); then, we define a shortfall of cumulative residual Tsallis (CRTES) and shortfall of right-tail deviation entropy (RTDS) and provide some equivalent results. As illustrated examples, the CRTESs of elliptical, inverse Gaussian, gamma and beta distributions are simulated.

A Generalized Measure of Cumulative Residual Entropy

In this work, we introduce a generalized measure of cumulative residual entropy and study its properties. We show that several existing measures of entropy, such as cumulative residual entropy, weighted cumulative residual entropy and cumulative residual Tsallis entropy, are all special cases of this generalized cumulative residual entropy. We also propose a measure of generalized cumulative entropy, which includes cumulative entropy, weighted cumulative entropy and cumulative Tsallis entropy as special cases. We discuss a generating function approach, using which we derive different entropy measures. We provide residual and cumulative versions of Sharma–Taneja–Mittal entropy and obtain them as special cases this generalized measure of entropy. Finally, using the newly introduced entropy measures, we establish some relationships between entropy and extropy measures.

Fractional Entropy-Based Test of Uniformity with Power Comparisons

In the present paper, we use the fractional and weighted cumulative residual entropy measures to test the uniformity. The limit distribution and an approximation of the distribution of the test statistic based on the fractional cumulative residual entropy are derived. Moreover, for this test statistic, percentage points and power against seven alternatives are reported. Finally, a simulation study is carried out to compare the power of the proposed tests and other tests of uniformity.

On Cumulative Entropies in Terms of Moments of Order Statistics

In this paper, relations between some kinds of cumulative entropies and moments of order statistics are established. By using some characterizations and the symmetry of a non-negative and absolutely continuous random variable X, lower and upper bounds for entropies are obtained and illustrative examples are given. By the relations with the moments of order statistics, a method is shown to compute an estimate of cumulative entropies and an application to testing whether data are exponentially distributed is outlined.

Connections between Weighted Generalized Cumulative Residual Entropy and Variance

A shift-dependent information measure is favorable to handle in some specific applied contexts such as mathematical neurobiology and survival analysis. For this reason, the weighted differential entropy has been introduced in the literature. In accordance with this measure, we propose the weighted generalized cumulative residual entropy as well. Despite existing apparent similarities between these measures, however, there are quite substantial and subtle differences between them because of their different metrics. In this paper, particularly, we show that the proposed measure is equivalent to the generalized cumulative residual entropy of the cumulative weighted random variable. Thus, we first provide expressions for the variance and the new measure in terms of the weighted mean residual life function and then elaborate on some characteristics of such measures, including equivalent expressions, stochastic comparisons, bounds, and connection with the excess wealth transform. Finally, we also illustrate some applications of interest in system reliability with reference to shock models and random minima.

(Generalized) Maximum Cumulative Direct, Residual, and Paired Φ 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 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.