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Varentropy of Past Lifetimes

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Abstract

In a variety of applicative fields the level of information in random quantities is commonly measured by means of the Shannon Entropy. In particular, in reliability theory and survival analysis, time-dependent generalizations of this measure of uncertainty have been considered to dynamically describe changes in the degree of information over time. The Residual Entropy and the Residual Varentropy, for example, have been considered in the specialized literature to measure the information and its variability in residual lifetimes. In a similar way, one can consider dynamic measures of information for past lifetimes, i.e., for random lifetimes of items when one assumes that their failures occur before a fixed inspection time. This paper provides a study of the Past Varentropy, defined as the dynamic measure of variability of information for past lifetimes. From this study emerges the interest on a particular family of lifetimes distributions, whose members satisfy the property to be the only ones having constant Past Varentropy.

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ACKNOWLEDGMENTS

The authors would like to thank the reviewers for their constructive comments that greatly improved the paper.

During the preparation of the final revised version of this paper, the authors have been informed that some of their results can also be found in the recent paper

– Raqab M.Z., Bayoud H.A., and Qiu G. (2022). ‘‘Varentropy of inactivity time of a random variable and its related applications,’’ IMA Journal of Mathematical Control and Information 39 (1), 132–154, which has already appeared but was submitted after the submission to this journal of this work.

The authors are members of the research group GNAMPA of INdAM (Istituto Nazionale di Alta Matematica). Francesco Buono and Maria Longobardi are partially supported by MIUR — PRIN 2017, project ‘‘Stochastic Models for Complex Systems’’, no. 2017 JFFHSH. The present work was developed within the activities of the project 000009_ALTRI_CDA_75_2021_FRA_LINEA_B funded by ‘‘Programma per il finanziamento della ricerca di Ateneo — Linea B’’ of the University of Naples Federico II.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni ‘‘Renato Caccioppoli’’, Università degli Studi di Napoli Federico II, I-80126, Napoli, Italy

    Francesco Buono

  2. Dipartimento di Biologia, Università degli Studi di Napoli Federico II, I-80126, Napoli, Italy

    Maria Longobardi

  3. Dipartimento di Scienze Matematiche, Politecnico di Torino, I-10129, Torino, Italy

    Franco Pellerey

Authors
  1. Francesco Buono
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  2. Maria Longobardi
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  3. Franco Pellerey
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Correspondence to Francesco Buono, Maria Longobardi or Franco Pellerey.

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Buono, F., Longobardi, M. & Pellerey, F. Varentropy of Past Lifetimes. Math. Meth. Stat. 31, 57–73 (2022). https://doi.org/10.3103/S106653072202003X

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  • DOI: https://doi.org/10.3103/S106653072202003X

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