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A copula-based approach to account for dependence in stress-strength models

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  • Filippo Domma
  • Sabrina Giordano

Abstract

The focus of stress-strength models is on the evaluation of the probability R = P(Y > X) that stress Y experienced by a component does not exceed strength X required to overcome it. In reliability studies, X and Y are typically modeled as independent. Nevertheless, in many applications such an assumption may be unrealistic. This is an interesting methodological issue, especially as the estimation of R for dependent stress and strength has received only limited attention to date. This paper aims to fill this gap by evaluating R taking into account the association between X and Y via a copula-based approach. We calculate a closed-form expression for R by modeling the dependence through a Farlie-Gumbel-Morgenstern copula and one of its extensions, numerical solutions for R are, instead, provided when members of Frank’s copula family are employed. The marginal distributions are assumed to belong to the Burr system (i.e. Burr III, Dagum or Singh-Maddala type). In all the cases, we prove that neglect of the existing dependence leads to higher or lower values of R than is the case. Copyright Springer-Verlag 2013

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  • Filippo Domma & Sabrina Giordano, 2013. "A copula-based approach to account for dependence in stress-strength models," Statistical Papers, Springer, vol. 54(3), pages 807-826, August.
    • Handle: RePEc:spr:stpapr:v:54:y:2013:i:3:p:807-826
    DOI: 10.1007/s00362-012-0463-0
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    References listed on IDEAS

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    7. Filippo Domma & Sabrina Giordano & Mariangela Zenga, 2011. "Maximum likelihood estimation in Dagum distribution with censored samples," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(12), pages 2971-2985, March.
    8. Filippo Domma & Sabrina Giordano, 2012. "A stress–strength model with dependent variables to measure household financial fragility," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(3), pages 375-389, August.
    9. Y. L. Lio & Tzong-Ru Tsai, 2012. "Estimation of δ= P ( X > Y ) for Burr XII distribution based on the progressively first failure-censored samples," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(2), pages 309-322, April.
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    Cited by:

    1. Nanami Taketomi & Kazuki Yamamoto & Christophe Chesneau & Takeshi Emura, 2022. "Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch," Mathematics, MDPI, vol. 10(20), pages 1-23, October.
    2. Takeshi Emura & Chi-Hung Pan, 2020. "Parametric likelihood inference and goodness-of-fit for dependently left-truncated data, a copula-based approach," Statistical Papers, Springer, vol. 61(1), pages 479-501, February.
    3. Fatih Kızılaslan & Mustafa Nadar, 2018. "Estimation of reliability in a multicomponent stress–strength model based on a bivariate Kumaraswamy distribution," Statistical Papers, Springer, vol. 59(1), pages 307-340, March.
    4. Alessandro Barbiero & Asmerilda Hitaj, 2022. "Approximation of continuous random variables for the evaluation of the reliability parameter of complex stress–strength models," Annals of Operations Research, Springer, vol. 315(2), pages 1573-1598, August.
    5. Shih, Jia-Han & Emura, Takeshi, 2021. "On the copula correlation ratio and its generalization," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    6. Jia-Han Shih & Takeshi Emura, 2018. "Likelihood-based inference for bivariate latent failure time models with competing risks under the generalized FGM copula," Computational Statistics, Springer, vol. 33(3), pages 1293-1323, September.
    7. Gijbels, Irène & Herrmann, Klaus, 2014. "On the distribution of sums of random variables with copula-induced dependence," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 27-44.
    8. A. James & N. Chandra & Nicy Sebastian, 2023. "Stress-strength reliability estimation for bivariate copula function with rayleigh marginals," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 14(1), pages 196-215, March.
    9. Jia-Han Shih & Takeshi Emura, 2019. "Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula," Statistical Papers, Springer, vol. 60(4), pages 1101-1118, August.
    10. Pushpa Narayan Rathie & Luan Carlos de Sena Monteiro Ozelim & Bernardo Borba de Andrade, 2021. "Portfolio Management of Copula-Dependent Assets Based on P ( Y < X ) Reliability Models: Revisiting Frank Copula and Dagum Distributions," Stats, MDPI, vol. 4(4), pages 1-24, December.
    11. Kosuke Nakazono & Yu-Cheng Lin & Gen-Yih Liao & Ryuji Uozumi & Takeshi Emura, 2024. "Computation of the Mann–Whitney Effect under Parametric Survival Copula Models," Mathematics, MDPI, vol. 12(10), pages 1-22, May.

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