Generalized Cumulative Residual Entropy for  Distributions with Unrestricted Supports

Drissi, Noomane; Chonavel, Thierry; Boucher, Jean Marc

doi:10.1155/2008/790607

Cited by 36 publications

(25 citation statements)

References 9 publications

(21 reference statements)

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“…Many other extensions of Shannon entropy are redefined by the idea in Rao et al (2004) via replacing f (x) by 1 − F(x); see, for example, the works done by Drissi et al (2008), Sunoj and Linu (2012), Psarrakos and Navarro (2013), Sati and Gupta (2015), Rajesh and Sunoj (2016) and Kundu et al (2016). Asadi and Zohrevand (2007) defined the dynamic version of the cumulative residual entropy (DCRE), which is the CRE of the residual random variable X t , and it is given by…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Some Characterization Results on Dynamic Cumulative Past Inaccuracy Measure

Khorashadizadeh

2018

JIRSS

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Abstract. In this paper, borrowing the intuition in Rao et al. (2004), we introduce a cumulative version of the inaccuracy measure (CIM). Also we obtain interesting and applicable properties of CIM for different cases based on the residual, past and interval lifetime random variables. Relying on various applications of stochastic classes in reliability and information theory fields, we study new classes of the lifetime in terms of the CIM along with their relations with other famous aging classes. Furthermore, some characterization results are obtained under the proportional reversed hazard rate model. Finally, considering that the time t changes in the range (t 1 , t 2 ), an extension of the CIM, called the interval cumulative residual (past) inaccuracy (ICR(P)I), is derived. We investigate the ICRI's relation with its analogous version based on Shannon entropy.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Some Characterization Results on Dynamic Cumulative Past Inaccuracy Measure

Khorashadizadeh

2018

JIRSS

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“…Moreover, Zografos et al [49] generalized the Shannon-type cumulative residual entropy to an entropy of the Rényi type. Furthermore, Drissi et al [50] considered random variables with general support. They also presented solutions for the maximization of Equation (3), provided that more general restrictions are considered.…”

Section: Reliability Theorymentioning

confidence: 99%

“…Following the arguments in [46,50], which were used for the special case of cumulative residual and residual Shannon entropy, one can derive stricter sufficient conditions for the existence of CPE α .…”

Section: Stricter Conditions For the Existence Of Cpe αmentioning

confidence: 99%

“…The cumulative residual entropy (CRE) introduced by [46], the generalized cumulative residual entropy (GCRE) of [50], and the cumulative entropy (CE) discussed by [8,9], have always been interpreted as measures of information. However, all these approaches do not explain which kind of information was considered.…”

Section: Basic Properties Of Cpe φmentioning

confidence: 99%

“…Rao et al [46] using the term "cross entropy", (p. 3) in [50]). MCPI φ is equivalent to the transinformation that is defined for Shannon's differential entropy (cf.…”

Section: Mutual Cumulative φ-Informationmentioning

confidence: 99%

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Cumulative Paired φ-Entropy

Klein

Mangold

Doll

2016

Entropy

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Abstract:A new kind of entropy will be introduced which generalizes both the differential entropy and the cumulative (residual) entropy. The generalization is twofold. First, we simultaneously define the entropy for cumulative distribution functions (cdfs) and survivor functions (sfs), instead of defining it separately for densities, cdfs, or sfs. Secondly, we consider a general "entropy generating function" φ, the same way Burbea et al. (IEEE Trans. Inf. Theory 1982, 28, 489-495) and Liese et al. (Convex Statistical Distances; Teubner-Verlag, 1987) did in the context of φ-divergences. Combining the ideas of φ-entropy and cumulative entropy leads to the new "cumulative paired φ-entropy" (CPE φ ). This new entropy has already been discussed in at least four scientific disciplines, be it with certain modifications or simplifications. In the fuzzy set theory, for example, cumulative paired φ-entropies were defined for membership functions, whereas in uncertainty and reliability theories some variations of CPE φ were recently considered as measures of information. With a single exception, the discussions in the scientific disciplines appear to be held independently of each other. We consider CPE φ for continuous cdfs and show that CPE φ is rather a measure of dispersion than a measure of information. In the first place, this will be demonstrated by deriving an upper bound which is determined by the standard deviation and by solving the maximum entropy problem under the restriction of a fixed variance. Next, this paper specifically shows that CPE φ satisfies the axioms of a dispersion measure. The corresponding dispersion functional can easily be estimated by an L-estimator, containing all its known asymptotic properties. CPE φ is the basis for several related concepts like mutual φ-information, φ-correlation, and φ-regression, which generalize Gini correlation and Gini regression. In addition, linear rank tests for scale that are based on the new entropy have been developed. We show that almost all known linear rank tests are special cases, and we introduce certain new tests. Moreover, formulas for different distributions and entropy calculations are presented for CPE φ if the cdf is available in a closed form.

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On Cumulative Entropies and Lifetime Estimations

Crescenzo

Longobardi

2009

Methods and Models in Artificial and Natural Computation. A Homage to Professor Mira’s Scientific Legacy

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Generalized Cumulative Residual Entropy for Distributions with Unrestricted Supports

Cited by 36 publications

References 9 publications

Some Characterization Results on Dynamic Cumulative Past Inaccuracy Measure

Some Characterization Results on Dynamic Cumulative Past Inaccuracy Measure

Cumulative Paired φ-Entropy

On Cumulative Entropies and Lifetime Estimations

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