Kybernetika 55 no. 3, 435-454, 2019

A generalized bivariate lifetime distribution based on parallel-series structures

Vahideh Mohtashami-Borzadaran, Mohammad Amini and Jafar AhmadiDOI: 10.14736/kyb-2019-3-0435

Abstract:

In this paper, a generalized bivariate lifetime distribution is introduced. This new model is constructed based on a dependent model consisting of two parallel-series systems which have a random number of parallel subsystems with fixed components connected in series. The probability that one system fails before the other one is measured by using competing risks. Using the extreme-value copulas, the dependence structure of the proposed model is studied. Kendall's tau, Spearman's rho and tail dependences are investigated for some special cases. Simulation results are given to examine the effectiveness of the proposed model.

Keywords:

copula, distortion, extreme-value copula, dependence measures, competing risks

Classification:

60E05, 62N05, 62H20

References:

  1. A. Asgharzadeh, H. S. Bakouch, S. Nadarajah and L. Esmaeili: A new family of compound lifetime distributions. Kybernetika 50 (2014), 142-169.   DOI:10.14736/kyb-2014-1-0142
  2. G. Crane and J. Hoek: Using distortions of copulas to price synthetic CDOs. Insurance Math. Econom. 42 (2008), 903-908.   DOI:10.1016/j.insmatheco.2007.10.007
  3. M. J. Crowder: Classical Competing Risks. Chapman and Hall/CRC, New York 2001.   DOI:10.1201/9781420035902
  4. A. Dolati, M. Amini and S. M. Mirhosseini: Dependence properties of bivariate distributions with proportional (reversed) hazards marginals. Metrika 77 (2014), 333-347.   DOI:10.1007/s00184-013-0440-1
  5. A. Dolati and M. Ućbeda-Flores: Constructing copulas by means of pairs of order statistics. Kybernetika 45 (2009), 992-1002.   CrossRef
  6. F. Durante: Construction of non-exchangeable bivariate distribution functions. Statist. Papers 50 (2009), 383-391.   DOI:10.1007/s00362-007-0064-5
  7. F. Durante, R. Foschi and P. Sarkoci: Distorted copulas: constructions and tail dependence. Commun. Statistics-Theory Methods 39 (2010), 2288-2301.   DOI:10.1080/03610920903039506
  8. V. Durrleman, A. Nickeghbali and T. Roncalli: A simple transformation of copulas. Available at: gro.creditlyonnais.fr/content/rd/home copulas.htm, (2000).   CrossRef
  9. C. Genest and L. Rivest: On the multivariate probability integral transformation. Statist. Probab. Lett. 53 (2001), 391-399.   DOI:10.1016/s0167-7152(01)00047-5
  10. M. Goldoust, S. Rezaei, Y. Si and S. Nadarajah: A lifetime distribution motivated by parallel and series structures. Commun. Statistics-Theory Methods 47 (2017), 3052-3072.   DOI:10.1080/03610926.2017.1346802
  11. G. Goudendorf and J. Segers: Extreme Value Copulas. In: Copula Theory and Its Applications, Lecture Notes in Statistics (P. Jaworski, F. Durante, W. Härdle, T. Rychlik, eds.), Springer, Berlin, Heidelberg 2010, 198, pp. 127-145.   DOI:10.1007/978-3-642-12465-5\_6
  12. H. Joe: Multivariate Models and Dependence Concepts. Chapman and Hall-London, Berlin 1997.   DOI:10.1002/(sici)1097-0258(19980930)17:18<2154::aid-sim913>3.0.co;2-r
  13. S. Kotz and S. Nadarajah: Extreme-Value Distributions: Theory and Applications. Imperial College Press-London, Berlin 2001.   DOI:10.1142/9781860944024
  14. S. Kotz, Y. Lumelskii and M. Pensky: The Stress-Strength Models and its Generalizations. World Scientific Publishing-Singapore, Berlin 2003.   DOI:10.1142/5015
  15. D. Kundu and A. Gupta: On bivariate Weibull-geometric distribution. J. Multivariate Anal. 123 (2014), 19-29.   DOI:10.1016/j.jmva.2013.08.004
  16. C. Li and X. Li: Preservation of increasing convex/concave order under the formation of parallel/series system of dependent components. Metrika 81 (2018), 4, 445-464.   DOI:10.1007/s00184-018-0651-6
  17. Y. Lu: The distribution of unobserved heterogeneity in competing risks models. Statist. Papers (2017).   DOI:10.1007/s00362-017-0956-y
  18. A. W. Marshall and I. Olkin: A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84 (1997), 641-652.   DOI:10.1093/biomet/84.3.641
  19. S. M. Mirhosseini, A. Dolati and M. Amini: On a class of distributions generated by stochastic mixture of the extreme order statistics of a sample of size two. J. Statist. Theory Appl. 10 (2011), 455-468.   CrossRef
  20. S. M. Mirhosseini, M. Amini and A. Dolati: On a general structure of the bivariate FGM type distributions. Appl. Math. 60 (2015), 91-108.   DOI:10.1007/s10492-015-0086-6
  21. P. Morillas: A method to obtain new copulas from a given one. Metrika 61 (2005), 169-184.   DOI:10.1007/s001840400330
  22. R. B. Nelsen: An Introduction to Copulas. Springer Science and Business Media, 2006.   DOI:10.1007/0-387-28678-0
  23. J. Pikhands: Multivariate extreme-value distributions (with a discussion). In: Proc. 43rd session of the International Statistical Institute. Bull. Int. Inst. 49 (1981), pp. 859-878.   CrossRef
  24. B. V. Popović and A. İ. Genç: On extremes of two-dimensional Student-t distribution of the Marshall-Olkin type. Mediter. J. Math. 15 (2018), 4, Article: 153.   DOI:10.1007/s00009-018-1201-1
  25. R. Roozegar and A. A. Jafari: On bivariate generalized linear failure rate-power series class of distributions. Iran. J. Science Technol., Trans. A: Science 41 (2017), 693-706.   DOI:10.1007/s40995-017-0297-7
  26. R. Roozegar and S. Nadarajah: New classes of power series bivariate copulas. J. Comput. Appl. Math. 326 (2017), 235-246.   DOI:10.1016/j.cam.2017.05.020
  27. J. H. Shih and T. Emura: Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula. Statist. Papers (2016).   DOI:10.1007/s00362-016-0865-5
  28. A. Sklar: Function de repartition an dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229-331.   CrossRef
  29. K. Zhang, J. Lin and P. Xu: A new class of copulas involving geometric distribution: Estimation and applications. Insurance: Mathematics and Economics 66 (2016), 1-10.   DOI:10.1016/j.insmatheco.2015.09.008