Journal of Sciences, Islamic Republic of Iran
A New Bivariate Shock Model Covering All Degrees of Dependencies
Document Type : Original Paper
Authors
Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Islamic Republic of Iran
Abstract
This paper presents a bivariate distribution that improves the Marshall-Olkin exponential shock model. The new construction method enhances the model’s capacity to include a common joint shock across components, making it especially suitable for reliability and credit risk assessments. The model features a single component and supports negative dependence structures. We investigate the key dependence properties and conduct a stress-strength analysis. After evaluating the performance of the parameter estimator, chemical engineering data is analyzed.
Keywords
Main Subjects
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18. Bayramoglu, I, and Ozkut, M. The reliability of coherent systems subjected to Marshall–Olkin type shocks. IEEE Transactions on Reliability. 2014;64(1):435–443.
19. Kundu, D, Franco, M, and Vivo, J-M. Multivariate distributions with proportional reversed hazard marginals. Computational Statistics and Data Analysis. 2014;77:98–112.
20. Cha, JH, and Badía, F. Multivariate reliability modeling based on dependent dynamic shock models. Applied Mathematical Modelling. 2017;51:199–216.
21. Al-Mutairi, D, Ghitany, M, and Kundu, D. Weighted Weibull distribution: Bivariate and multivariate cases. Brazilian Journal of Probability and Statistics. 2018;32(1):20–43.
22. Mohtashami-Borzadaran, HA, Jabbari, H, and Amini, M. Bivariate Marshall–Olkin exponential shock model. Probability in the Engineering and Informational Sciences. 2021;35(3):745–765.
23. El Ktaibi, F, Bentoumi, R, Sottocornola, N, and Mesfioui, M. Bivariate copulas based on counter-monotonic shock method. Risks. 2022; 10:202–222.
24. Bentoumi, R, El Ktaibi, F, and Mesfioui, M. A new family of bivariate exponential distributions with negative dependence based on counter-monotonic shock method. Entropy. 2021;23:548–565.
25. Lee, H, and Cha, JH. New continuous bivariate distributions generated from shock models. Statistics. 2024;58(2):437–449.
26. Agrawal, A, Ganguly, A, and Mitra, D. Modelling competing risks data by a bivariate model with singularity originating from a shock model using the Lehmann family of distributions. Research in Statistics. 2025;3(1):1–13.
27. Genest, C, Mesfioui, M, and Schulz, J. A new bivariate Poisson common shock model covering all possible degrees of dependence. Statistics and Probability Letters. 2018;140:202–209.
28. Mohtashami-Borzadaran, HA, Amini, M, Jabbari, H, and Dolati, A. Marshall-Olkin exponential shock model covering all ranges of dependence. arXiv preprint arXiv: 2004.11241. 2020.
29. Joe, H. Multivariate Models and Multivariate Dependence Concepts. Chapman and Hall/CRC, New York. 1997.
30. Nelsen, RB. An Introduction to Copulas. Springer-Verlag, New York, 2007.
31. Ghalanos, A, Theussl, S, and Ghalanos, MA. Package "Rsolnp". 2012.
32. Mohsin, M, Kazianka, H, Pilz, J, and Gebhardt, A. A new bivariate exponential distribution for modeling moderately negative dependence. Statistical Methods and Applications. 2014;23(1):123–148.
33. Lange, TR, Royals, H, and Connor, LL. Influence of water chemistry on mercury concentration in largemouth bass from Florida lakes. Transactions of the American Fisheries Society. 1993;122(1):74–84.
2. Lai, CD, and Balakrishnan, N. Continuous Bivariate Distributions. Springer-Verlag, New York. 2009.
3. Marshall, AW, and Olkin, I. A multivariate exponential distribution. Journal of the American Statistical Association. 1967;62(317):30–44.
4. Cherubini, U, Durante, F, and Mulinacci, S. Marshall-Olkin distributions. in Advances in Theory and Applications. Springer Proceedings in Mathematics and Statistics, Springer. 2015.
5. Lindskog, F, and McNeil, A.J. Common Poisson shock models: Applications to insurance and credit risk modeling. ASTIN Bulletin: The Journal of the IAA. 2003;33(2):209–238.
6. Cherubini, U, and Mulinacci, S. The Gumbel-Marshall-Olkin distribution. In Copulas and Dependence Models with Applications, Springer. 2017.
7. Esary, JD, and Marshall, AW. Multivariate distributions with exponential minimums. The Annals of Statistics. 1974;2(1):84–98.
8. Raftery, AE. A continuous multivariate exponential distribution. Communications in Statistics-Theory and Methods. 1984;13(8):947–965.
9. Tawn, JA. Modeling multivariate extreme value distributions. Biometrika. 1990;77(2):245–253.
10. Lin, H-H, Chen, K, and Wang, R-T. A multivariant exponential shared-load model. IEEE Transactions on Reliability. 1993;42(1):165–171.
11. Basu, AP, and Sun, K. Multivariate exponential distributions with constant failure rates. Journal of Multivariate Analysis. 1997;61(2):159–170.
12. Gómez, E, Gomez-Viilegas, M, and Marin, J. A multivariate generalization of the power exponential family of distributions. Communications in Statistics-Theory and Methods. 1998;27(3):589–600.
13. Cui, L, and Li, H. Analytical method for reliability and MTTF assessment of coherent systems with dependent components. Reliability Engineering and System Safety. 2007;92(3):300–307.
14. Fan, J, Nunn, ME, and Su, X. Multivariate exponential survival trees and their application to tooth prognosis. Computational Statistics and Data Analysis. 2009;53(4):1110–1121.
15. Kundu, D, and Gupta, RD. Bivariate generalized exponential distribution. Journal of Multivariate Analysis. 2009;100(4):581–593.
16. Li, X, and Pellerey, F. Generalized Marshall–Olkin distributions and related bivariate aging properties. Journal of Multivariate Analysis. 2011;102(10):1399–1409.
17. Kundu, D, and Gupta, AK. Bayes estimation for the Marshall–Olkin bivariate Weibull distribution. Computational Statistics and Data Analysis. 2013;57(1):271–281.
18. Bayramoglu, I, and Ozkut, M. The reliability of coherent systems subjected to Marshall–Olkin type shocks. IEEE Transactions on Reliability. 2014;64(1):435–443.
19. Kundu, D, Franco, M, and Vivo, J-M. Multivariate distributions with proportional reversed hazard marginals. Computational Statistics and Data Analysis. 2014;77:98–112.
20. Cha, JH, and Badía, F. Multivariate reliability modeling based on dependent dynamic shock models. Applied Mathematical Modelling. 2017;51:199–216.
21. Al-Mutairi, D, Ghitany, M, and Kundu, D. Weighted Weibull distribution: Bivariate and multivariate cases. Brazilian Journal of Probability and Statistics. 2018;32(1):20–43.
22. Mohtashami-Borzadaran, HA, Jabbari, H, and Amini, M. Bivariate Marshall–Olkin exponential shock model. Probability in the Engineering and Informational Sciences. 2021;35(3):745–765.
23. El Ktaibi, F, Bentoumi, R, Sottocornola, N, and Mesfioui, M. Bivariate copulas based on counter-monotonic shock method. Risks. 2022; 10:202–222.
24. Bentoumi, R, El Ktaibi, F, and Mesfioui, M. A new family of bivariate exponential distributions with negative dependence based on counter-monotonic shock method. Entropy. 2021;23:548–565.
25. Lee, H, and Cha, JH. New continuous bivariate distributions generated from shock models. Statistics. 2024;58(2):437–449.
26. Agrawal, A, Ganguly, A, and Mitra, D. Modelling competing risks data by a bivariate model with singularity originating from a shock model using the Lehmann family of distributions. Research in Statistics. 2025;3(1):1–13.
27. Genest, C, Mesfioui, M, and Schulz, J. A new bivariate Poisson common shock model covering all possible degrees of dependence. Statistics and Probability Letters. 2018;140:202–209.
28. Mohtashami-Borzadaran, HA, Amini, M, Jabbari, H, and Dolati, A. Marshall-Olkin exponential shock model covering all ranges of dependence. arXiv preprint arXiv: 2004.11241. 2020.
29. Joe, H. Multivariate Models and Multivariate Dependence Concepts. Chapman and Hall/CRC, New York. 1997.
30. Nelsen, RB. An Introduction to Copulas. Springer-Verlag, New York, 2007.
31. Ghalanos, A, Theussl, S, and Ghalanos, MA. Package "Rsolnp". 2012.
32. Mohsin, M, Kazianka, H, Pilz, J, and Gebhardt, A. A new bivariate exponential distribution for modeling moderately negative dependence. Statistical Methods and Applications. 2014;23(1):123–148.
33. Lange, TR, Royals, H, and Connor, LL. Influence of water chemistry on mercury concentration in largemouth bass from Florida lakes. Transactions of the American Fisheries Society. 1993;122(1):74–84.
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