Fractional generalized cumulative entropy and its dynamic version

•A fractional version of the generalized cumulative entropy is introduced.•The proposed notion is a variability measure connected with fractional integrals.•This is suitable for distributions satisfying the proportional reversed hazard model.•The related dynamic measure is an extension of the mean i...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation Vol. 102; p. 105899
Main Authors: Di Crescenzo, Antonio, Kayal, Suchandan, Meoli, Alessandra
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.11.2021
Elsevier Science Ltd
Subjects:
Cumulative entropy
Distribution functions
Entropy
Estimation
Fractional calculus
Parameter estimation
Probability distribution functions
Stochastic models
Stochastic orderings
Stochastic orderings
Cumulative entropy
Estimation
Fractional calculus
ISSN:1007-5704, 1878-7274
Online Access:Get full text
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Summary:•A fractional version of the generalized cumulative entropy is introduced.•The proposed notion is a variability measure connected with fractional integrals.•This is suitable for distributions satisfying the proportional reversed hazard model.•The related dynamic measure is an extension of the mean inactivity time.•We discuss properties of the empirical measure and apply it to real data. Following the theory of information measures based on the cumulative distribution function, we propose the fractional generalized cumulative entropy, and its dynamic version. These entropies are particularly suitable to deal with distributions satisfying the proportional reversed hazard model. We study the connection with fractional integrals, and some bounds and comparisons based on stochastic orderings, that allow to show that the proposed measure is actually a variability measure. The investigation also involves various notions of reliability theory, since the considered dynamic measure is a suitable extension of the mean inactivity time. We also introduce the empirical generalized fractional cumulative entropy as a non-parametric estimator of the new measure. It is shown that the empirical measure converges to the proposed notion almost surely. Then, we address the stability of the empirical measure and provide an example of application to real data. Finally, a central limit theorem is established under the exponential distribution.
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ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2021.105899