IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i15p1782-d603076.html
   My bibliography  Save this article

Residual Probability Function for Dependent Lifetimes

Author

Listed:
  • Mhamed Mesfioui

    (Département de Mathématiques et D’informatique, Université du Québec à Trois-Rivières, 3351 Boulevard des Forges, Trois-Rivières, QC G9A 5H7, Canada)

  • Mohamed Kayid

    (Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

Abstract

In this paper, the residual probability function is applied to analyze the survival probability of two used components relative to each other in the case when their lifetimes are dependent. The expression of the function by copulas has been derived along with some examples of particular copulas. The behaviour of the residual probability function in terms of the underlying dependence is also discussed. The residual probability order is also considered in the dependent case. In the class of Archimedean survival copulas, we prove that the residual probability order implies the usual stochastic order in the reversed direction, and the hazard rate order concludes the residual probability order.

Suggested Citation

  • Mhamed Mesfioui & Mohamed Kayid, 2021. "Residual Probability Function for Dependent Lifetimes," Mathematics, MDPI, vol. 9(15), pages 1-13, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1782-:d:603076
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/15/1782/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/15/1782/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. M. Kayid & S. Izadkhah & S. Alshami, 2014. "Residual Probability Function, Associated Orderings, and Related Aging Classes," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-10, December.
    2. Bassan, Bruno & Spizzichino, Fabio, 2005. "Bivariate survival models with Clayton aging functions," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 6-12, August.
    3. V. Zardasht & M. Asadi, 2010. "Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(4), pages 460-481, November.
    4. M. Kayid, 2011. "Preservation properties of the moment generating function ordering of residual lives," Statistical Papers, Springer, vol. 52(3), pages 523-529, August.
    5. Maxim Finkelstein & Veronica Esaulova, 2005. "On the weak IFR aging of bivariate lifetime distributions," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 21(3), pages 265-272, May.
    6. Sabrina Mulinacci, 2018. "Archimedean-based Marshall-Olkin Distributions and Related Dependence Structures," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 205-236, March.
    7. Gupta, Nitin & Misra, Neeraj & Kumar, Somesh, 2015. "Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components," European Journal of Operational Research, Elsevier, vol. 240(2), pages 425-430.
    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. 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.
    2. Jørgen Vitting Andersen & Roy Cerqueti & Giulia Rotundo, 2017. "Rational expectations and stochastic systems," Documents de travail du Centre d'Economie de la Sorbonne 17060, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Oct 2019.
    3. Jorgen Vitting Andersen & Roy Cerqueti & Jessica Riccioni, 2021. "Rational expectations as a tool for predicting failure of weighted k-out-of-n reliability systems," Papers 2112.10672, arXiv.org.
    4. Kokol Bukovšek, Damjana & Košir, Tomaž & Mojškerc, Blaž & Omladič, Matjaž, 2022. "Extreme generators of shock induced copulas," Applied Mathematics and Computation, Elsevier, vol. 429(C).
    5. Umberto Cherubini & Sabrina Mulinacci, 2021. "Hierarchical Archimedean Dependence in Common Shock Models," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 143-163, March.
    6. Davies, Katherine & Dembińska, Anna, 2024. "On the residual lifetimes of dependent components upon system failure," Reliability Engineering and System Safety, Elsevier, vol. 248(C).
    7. Parsa, Motahareh & Di Crescenzo, Antonio & Jabbari, Hadi, 2018. "Analysis of reliability systems via Gini-type index," European Journal of Operational Research, Elsevier, vol. 264(1), pages 340-353.
    8. Sabrina Mulinacci, 2022. "A Marshall-Olkin Type Multivariate Model with Underlying Dependent Shocks," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2455-2484, December.
    9. Iain L. MacDonald, 2021. "Is EM really necessary here? Examples where it seems simpler not to use EM," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(4), pages 629-647, December.
    10. Zhu, Xiaojun & Balakrishnan, N., 2023. "Non-parametric inference based on reliability life-test of non-identical coherent systems with application to warranty time," Reliability Engineering and System Safety, Elsevier, vol. 232(C).
    11. Mulero, Julio & Pellerey, Franco & Rodríguez-Griñolo, Rosario, 2010. "Stochastic comparisons for time transformed exponential models," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 328-333, April.
    12. Charpentier, Arthur & Segers, Johan, 2007. "Lower tail dependence for Archimedean copulas: Characterizations and pitfalls," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 525-532, May.
    13. Félix Belzunce & Carolina Martínez-Riquelme, 2019. "On the unimodality of the likelihood ratio with applications," Statistical Papers, Springer, vol. 60(1), pages 223-237, February.
    14. Rychlik, Tomasz, 2017. "Evaluations of quantiles of system lifetime distributions," European Journal of Operational Research, Elsevier, vol. 256(3), pages 935-944.
    15. Salehi, Ebrahim & Tavangar, Mahdi, 2019. "Stochastic comparisons on conditional residual lifetime and inactivity time of coherent systems with exchangeable components," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 327-337.
    16. Jorge Navarro, 2018. "Distribution-free comparisons of residual lifetimes of coherent systems based on copula properties," Statistical Papers, Springer, vol. 59(2), pages 781-800, June.
    17. Chen Li & Xiaohu Li, 2020. "Weak aging properties for coherent systems with statistically dependent component lifetimes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(7), pages 559-572, October.
    18. Jørgen Vitting Andersen & Roy Cerqueti & Jessica Riccioni, 2023. "Rational expectations as a tool for predicting failure of weighted k-out-of-n reliability systems," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-03634370, HAL.
    19. Jørgen Vitting Andersen & Roy Cerqueti & Jessica Riccioni, 2023. "Rational expectations as a tool for predicting failure of weighted k-out-of-n reliability systems," Post-Print hal-03634370, HAL.
    20. Charpentier, A. & Segers, J.J.J., 2006. "Lower Tail Dependence for Archimedean Copulas : Characterizations and Pitfalls," Other publications TiSEM ae669e5a-1929-42d9-b137-6, Tilburg University, School of Economics and Management.

    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:gam:jmathe:v:9:y:2021:i:15:p:1782-:d:603076. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.