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Distribution-free comparisons of residual lifetimes of coherent systems based on copula properties

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  • Jorge Navarro

    (Universidad de Murcia)

Abstract

In this paper a general procedure is proposed to get stochastic comparisons of residual lifetimes of coherent systems. The comparison results obtained are based on the structure of the system and on properties of the copula used to describe the dependence between the component lifetimes. They are distribution-free with respect to the component lifetime distributions. We consider two system residual lifetimes at a given time $$t>0$$ t > 0 . In the first one, we just assume that the system is working at time t while, in the second one, we assume that all the components are working at time t. We study the main stochastic orderings: hazard rate, stochastic, mean residual life and likelihood ratio orders. Specific results are obtained in the particular case of independent components and in the case of identically distributed components. Some illustrative examples are included. They prove that, in some cases, these system residual lifetimes are not ordered. Even more, surprisingly, sometimes the first residual lifetime is greater (in some stochastic sense) than the second one under a given dependence model.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:59:y:2018:i:2:d:10.1007_s00362-016-0789-0
    DOI: 10.1007/s00362-016-0789-0
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    References listed on IDEAS

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    1. Navarro, Jorge & Rychlik, Tomasz, 2010. "Comparisons and bounds for expected lifetimes of reliability systems," European Journal of Operational Research, Elsevier, vol. 207(1), pages 309-317, November.
    2. Jorge Navarro & M. Carmen Gomis, 2016. "Comparisons in the mean residual life order of coherent systems with identically distributed components," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 32(1), pages 33-47, January.
    3. Jorge Navarro, 2016. "Stochastic comparisons of generalized mixtures and coherent systems," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 150-169, March.
    4. Jorge Navarro & Pedro Hernandez, 2008. "Mean residual life functions of finite mixtures, order statistics and coherent systems," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 67(3), pages 277-298, April.
    5. Navarro, Jorge & Pellerey, Franco & Di Crescenzo, Antonio, 2015. "Orderings of coherent systems with randomized dependent components," European Journal of Operational Research, Elsevier, vol. 240(1), pages 127-139.
    6. Jorge Navarro & Fabio Spizzichino, 2010. "Comparisons of series and parallel systems with components sharing the same copula," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 26(6), pages 775-791, November.
    7. S. Gurler, 2012. "On residual lifetimes in sequential (n − k + 1)-out-of-n systems," Statistical Papers, Springer, vol. 53(1), pages 23-31, February.
    8. Jorge Navarro & Yolanda Águila & Miguel A. Sordo & Alfonso Suárez-Llorens, 2016. "Preservation of Stochastic Orders under the Formation of Generalized Distorted Distributions. Applications to Coherent Systems," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 529-545, June.
    9. 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.
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    Citations

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    Cited by:

    1. Murat Ozkut, 2022. "Comparison of the replacement policy in k-out-of-n systems having dependent components," Journal of Risk and Reliability, , vol. 236(1), pages 125-137, February.
    2. Eryilmaz, Serkan & Ozkut, Murat, 2020. "Optimization problems for a parallel system with multiple types of dependent components," Reliability Engineering and System Safety, Elsevier, vol. 199(C).
    3. Navarro, Jorge & Arriaza, Antonio & Suárez-Llorens, Alfonso, 2019. "Minimal repair of failed components in coherent systems," European Journal of Operational Research, Elsevier, vol. 279(3), pages 951-964.
    4. Sareh Goli, 2019. "On the conditional residual lifetime of coherent systems under double regularly checking," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(4), pages 352-363, June.
    5. Li, Chen & Li, Xiaohu, 2023. "On k-out-of-n systems with homogeneous components and one independent cold standby redundancy," Statistics & Probability Letters, Elsevier, vol. 203(C).
    6. repec:bpj:demode:v:6:y:2018:i:1:p:156-177:n:10 is not listed on IDEAS
    7. Torrado, Nuria & Arriaza, Antonio & Navarro, Jorge, 2021. "A study on multi-level redundancy allocation in coherent systems formed by modules," Reliability Engineering and System Safety, Elsevier, vol. 213(C).
    8. Jorge Navarro & Camilla Calì & Maria Longobardi & Fabrizio Durante, 2022. "Distortion representations of multivariate distributions," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(4), pages 925-954, October.

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