Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-17T09:06:31.155Z Has data issue: false hasContentIssue false
  • English
  • Français

STOCHASTIC ORDERING OF LIFETIMES OF PARALLEL AND SERIES SYSTEMS COMPRISING HETEROGENEOUS DEPENDENT COMPONENTS WITH GENERALIZED BIRNBAUM-SAUNDERS DISTRIBUTIONS

Published online by Cambridge University Press:  25 August 2020

Mehdi Amiri Open the ORCID record for Mehdi Amiri [Opens in a new window] ,
Narayanaswamy Balakrishnan  and
Ahad Jamalizadeh
Mehdi Amiri
Affiliation:
Department of Statistics, Faculty of Basic Sciences, University of Hormozgan, Bandarabbas, Iran E-mail: m.amiri@hormozgan.ac.ir
Narayanaswamy Balakrishnan
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada
Ahad Jamalizadeh
Affiliation:
Department of Statistics, Faculty of Mathematics & Computer, Shahid Bahonar University of Kerman, Kerman, Iran Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we discuss stochastic orderings of lifetimes of two heterogeneous parallel and series systems with heterogeneous dependent components having generalized Birnbaum–Saunders distributions. The comparisons presented here are based on the vector majorization of parameters. The ordering results are established in some special cases for the generalized Birnbaum–Saunders distribution based on the multivariate elliptical, normal, t, logistic, and skew-normal kernels. Further, we use these results by considering Archimedean copulas to model the dependence structure among systems with generalized Birnbaum–Saunders components. These results have been used to derive some upper and lower bounds for survival functions of lifetimes of parallel and series systems.

Keywords

Archimedean copulalog-concave functionmajorizationparallel systemSchur-concave functionseries systemstochastic order
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 1 , January 2022 , pp. 49 - 65
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12(2): 171–178.Google Scholar
Azzalini, A. & Capitanio, A. (1999). Statistical applications of the multivariate skew-normal distribution. Journal of the Royal Statistical Society, Series B 61(3): 579–602.CrossRefGoogle Scholar
Azzalini, A. & Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika 83(4): 715–726.CrossRefGoogle Scholar
Balakrishnan, N. & Kundu, D. (2019). Birnbaum-Saunders distribution: a review of models, analysis, and applications. Applied Stochastic Models in Business and Industry 35(1): 4–132 (with discussions).CrossRefGoogle Scholar
Balakrishnan, N. & Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences 27(4): 403–443.CrossRefGoogle Scholar
Balakrishnan, N., Haidari, A., & Masoumifard, K. (2015). Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Transactions on Reliability 64(1): 333–348.CrossRefGoogle Scholar
Birnbaum, Z.W. & Saunders, S.C. (1969). Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability 6(2): 328–347.CrossRefGoogle Scholar
Birnbaum, Z.W. & Saunders, S.C. (1969). A new family of life distributions. Journal of Applied Probability 6(2): 319–327.CrossRefGoogle Scholar
Díaz-García, J.A. & Leiva-Sánchez, V. (2005). A new family of life distributions based on the elliptically contoured distributions. Journal of Statistical Planning and Inference 128(2): 445–457.CrossRefGoogle Scholar
Dykstra, R., Kochar, S., & Rojo, J. (1997). Stochastic comparisons of parallel systems of heterogeneous exponential components. Journal of Statistical Planning and Inference 65(2): 203–211.CrossRefGoogle Scholar
Fang, K.T. & Anderson, T.W. (eds.) (1990). Statistical inference in elliptically contoured and related distributions. New York: Allerton Press.Google Scholar
Fang, L. & Balakrishnan, N. (2018). Ordering properties of the smallest order statistics from generalized Birnbaum-Saunders models with associated random shocks. Metrika 81(1): 19–35.CrossRefGoogle Scholar
Fang, L. & Zhu, X. (2019). Discussion of Birnbaum-Saunders distribution: a review of models, analysis and applications. Applied Stochastic Models in Business and Industry 35: 72–76.CrossRefGoogle Scholar
Fang, K.T., Kotz, S., & Ng, K.W (1990). Symmetric multivariate and related distributions. New York: Chapman and Hall.CrossRefGoogle Scholar
Fang, L., Zhu, X., & Balakrishnan, N. (2016). Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum-Saunders components. Statistics & Probability Letters 112: 131–136.CrossRefGoogle Scholar
Fang, L., Zhu, X., & Balakrishnan, N. (2018). Stochastic ordering of minima and maxima from heterogeneous bivariate Birnbaum–Saunders random vectors. Statistics 52(1): 147–155.CrossRefGoogle Scholar
Gupta, S.D., Eaton, M.L., Olkin, I., Perlman, M., Savage, L.J., & Sobel, M (1972). Inequalities on the probability content of convex regions for elliptically contoured distributions. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), vol. 2, pp. 241–265.Google Scholar
Jamalizadeh, A. & Kundu, D. (2015). A multivariate Birnbaum-Saunders distribution based on the multivariate skew normal distribution. Journal of the Japan Statistical Society 45(1): 1–20.CrossRefGoogle Scholar
Khaledi, B.E. & Kochar, S. (2000). Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability 37(4): 1123–1128.CrossRefGoogle Scholar
Khaledi, B.E. & Kochar, S. (2006). Weibull distribution: some stochastic comparisons results. Journal of Statistical Planning and Inference 136(9): 3121–3129.CrossRefGoogle Scholar
Kundu, D., Balakrishnan, N., & Jamalizadeh, A. (2010). Bivariate Birnbaum-Saunders distribution and associated inference. Journal of Multivariate Analysis 101(1): 113–125.CrossRefGoogle Scholar
Kundu, D., Balakrishnan, N., & Jamalizadeh, A. (2013). Generalized multivariate Birnbaum-Saunders distributions and related inferential issues. Journal of Multivariate Analysis 116: 230–244.CrossRefGoogle Scholar
Li, C. & Li, X. (2015). Likelihood ratio order of sample minimum from heterogeneous Weibull random variables. Statistics & Probability Letters 97: 46–53.CrossRefGoogle Scholar
Marshall, A.W., Olkin, I., & Arnold, B.C (2011). Inequalities: theory of majorization and its applications. New York: Springer.CrossRefGoogle Scholar
McNeil, A.J. & Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and $\ell _1$-norm symmetric distributions. The Annals of Statistics 37(5B): 3059–3097.CrossRefGoogle Scholar
Peajcariaac, J.E. & Tong, Y.L (1992). Convex functions, partial orderings, and statistical applications. New York: Academic Press.Google Scholar
Shaked, M. & Shanthikumar, G (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
Tong, Y.L (2012). The multivariate normal distribution. New York: Springer.Google Scholar
Torrado, N. & Kochar, S.C. (2015). Stochastic order relations among parallel systems from Weibull distributions. Journal of Applied Probability 52(1): 102–116.CrossRefGoogle Scholar
Vilca, F. & Leiva, V. (2006). A new fatigue life model based on the family of skew elliptical distributions. Communications in Statistics – Theory and Methods 35: 229–244.CrossRefGoogle Scholar