IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v24y2022i3d10.1007_s11009-021-09905-2.html
   My bibliography  Save this article

Multivariate Reversed Hazard Rates and Inactivity Times of Systems

Author

Listed:
  • Francesco Buono

    (Università degli Studi di Napoli Federico II)

  • Emilio Santis

    (Università La Sapienza)

  • Maria Longobardi

    (Università degli Studi di Napoli Federico II)

  • Fabio Spizzichino

    (Università La Sapienza)

Abstract

The family of the multivariate conditional hazard rate functions often reveals to be a convenient tool to describe the joint probability distribution of a vector of non-negative random variables (lifetimes) in the absolutely continuous case. Such a tool can have in particular an important role in the study of the behavior of the minima among inter-dependent lifetimes. In this paper we introduce the concept of reversed multivariate conditional hazard rate functions, which extends the one-dimensional notion of reversed hazard rate of a single non-negative random variable. Several basic properties of this concept are proven. In particular, we point out a related role in the study of the behavior of the maximum value among inter-dependent lifetimes. In different applied fields, and in particular in the reliability literature, a remarkable class of dependence models for vectors of lifetimes is related with the load-sharing condition, which can be defined in terms of the multivariate conditional hazard rate functions. In the paper we define the class of reversed load-sharing models, which can be seen as natural extensions to the multivariate case of the univariate inverse exponential distributions. We analyze basic properties of such a class of dependence models. In particular we show a result related to the study of the inactivity time of a coherent system when the joint distribution of the components’ lifetimes is a reversed load-sharing model.

Suggested Citation

  • Francesco Buono & Emilio Santis & Maria Longobardi & Fabio Spizzichino, 2022. "Multivariate Reversed Hazard Rates and Inactivity Times of Systems," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1987-2008, September.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09905-2
    DOI: 10.1007/s11009-021-09905-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-021-09905-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-021-09905-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Erhard Cramer & Udo Kamps, 2003. "Marginal distributions of sequential and generalized order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 58(3), pages 293-310, December.
    2. Moshe Shaked & J. George Shanthikumar, 1990. "Multivariate Stochastic Orderings and Positive Dependence in Reliability Theory," Mathematics of Operations Research, INFORMS, vol. 15(3), pages 545-552, August.
    3. Jorge Navarro & Maria Longobardi & Franco Pellerey, 2017. "Comparison results for inactivity times of k-out-of-n and general coherent systems with dependent components," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(4), pages 822-846, December.
    4. Moshe Shaked & J. George Shanthikumar, 2015. "Multivariate conditional hazard rate functions – an overview," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 31(3), pages 285-296, May.
    5. Belzunce, Félix & Mercader, José-Angel & Ruiz, José-María & Spizzichino, Fabio, 2009. "Stochastic comparisons of multivariate mixture models," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1657-1669, September.
    6. Francisco J. Samaniego, 2007. "System Signatures and their Applications in Engineering Reliability," International Series in Operations Research and Management Science, Springer, number 978-0-387-71797-5, April.
    7. Li, Xiaohu & Da, Gaofeng, 2010. "Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 1016-1025, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Foschi Rachele & Nappo Giovanna & Spizzichino Fabio L., 2021. "Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes," Dependence Modeling, De Gruyter, vol. 9(1), pages 394-423, January.
    2. Bezgina, E. & Burkschat, M., 2019. "On total positivity of exchangeable random variables obtained by symmetrization, with applications to failure-dependent lifetimes," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 95-109.
    3. Khaledi, Baha-Eldin & Shaked, Moshe, 2010. "Stochastic comparisons of multivariate mixtures," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2486-2498, November.
    4. Erhard Cramer & Jorge Navarro, 2015. "Progressive Type‐II censoring and coherent systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(6), pages 512-530, September.
    5. M. Burkschat & J. Navarro, 2014. "Asymptotic behavior of the hazard rate in systems based on sequential order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(8), pages 965-994, November.
    6. Badía, F.G. & Sangüesa, C. & Cha, J.H., 2014. "Stochastic comparison of multivariate conditionally dependent mixtures," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 82-94.
    7. Balakrishnan, Narayanaswamy & Barmalzan, Ghobad & Haidari, Abedin, 2016. "Multivariate stochastic comparisons of multivariate mixture models and their applications," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 37-43.
    8. Serkan Eryilmaz & Frank P.A. Coolen & Tahani Coolen‐Maturi, 2018. "Mean residual life of coherent systems consisting of multiple types of dependent components," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(1), pages 86-97, February.
    9. Somayeh Zarezadeh & Somayeh Ashrafi & Majid Asadi, 2018. "Network Reliability Modeling Based on a Geometric Counting Process," Mathematics, MDPI, vol. 6(10), pages 1-17, October.
    10. Asadi, Majid & Ebrahimi, Nader & Soofi, Ehsan S., 2018. "Optimal hazard models based on partial information," European Journal of Operational Research, Elsevier, vol. 270(2), pages 723-733.
    11. Xiaohu Li & Gaofeng Da & Peng Zhao, 2012. "Competing Between Two Groups of Individuals Following Frailty Models," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 1033-1051, December.
    12. Mansour Shrahili & Mohamed Kayid, 2023. "Stochastic Orderings of the Idle Time of Inactive Standby Systems," Mathematics, MDPI, vol. 11(20), pages 1-21, October.
    13. Gaofeng Da & Lvyu Xia & Taizhong Hu, 2014. "On Computing Signatures of k-out-of-n Systems Consisting of Modules," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 223-233, March.
    14. Marco Burkschat & Tomasz Rychlik, 2018. "Sharp inequalities for quantiles of system lifetime distributions from failure-dependent proportional hazard model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 618-638, September.
    15. Feng, Geng & Patelli, Edoardo & Beer, Michael & Coolen, Frank P.A., 2016. "Imprecise system reliability and component importance based on survival signature," Reliability Engineering and System Safety, Elsevier, vol. 150(C), pages 116-125.
    16. Zarezadeh, S. & Mohammadi, L. & Balakrishnan, N., 2018. "On the joint signature of several coherent systems with some shared components," European Journal of Operational Research, Elsevier, vol. 264(3), pages 1092-1100.
    17. Emilio De Santis & Yaakov Malinovsky & Fabio Spizzichino, 2021. "Stochastic Precedence and Minima Among Dependent Variables," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 187-205, March.
    18. Hindolo George-Williams & Geng Feng & Frank PA Coolen & Michael Beer & Edoardo Patelli, 2019. "Extending the survival signature paradigm to complex systems with non-repairable dependent failures," Journal of Risk and Reliability, , vol. 233(4), pages 505-519, August.
    19. Krzysztof Jasiński, 2021. "The number of failed components in a coherent working system when the lifetimes are discretely distributed," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(7), pages 1081-1094, October.
    20. van Bentum, Thomas & Cramer, Erhard, 2019. "Stochastic monotonicity of MLEs of the mean for exponentially distributed lifetimes under hybrid censoring," Statistics & Probability Letters, Elsevier, vol. 148(C), pages 1-8.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09905-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.