Skip to main content
SpringerLink
Log in

Reliability of a Multicomponent Stress-strength Model Based on a Bivariate Kumaraswamy Distribution with Censored Data

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we consider a system which has k statistically independent and identically distributed strength components and each component is constructed by a pair of statistically dependent elements with doubly type-II censored scheme. These elements (X1, Y1), (X2, Y2), ⋯, (Xk, Yk) follow a bivariate Kumaraswamy distribution and each element is exposed to a common random stress T which follows a Kumaraswamy distribution. The system is regarded as operating only if at least s out of k (1 ≤ s ≤ k) strength variables exceed the random stress. The multicomponent reliability of the system is given by Rs,k=P(at least s of the (Z1, ⋯, Zk) exceed T) where Zi = min(Xi, Yi), i = 1, ⋯, k. The Bayes estimates of Rs,k have been developed by using the Markov Chain Monte Carlo methods due to the lack of explicit forms. The uniformly minimum variance unbiased and exact Bayes estimates of Rs,k are obtained analytically when the common second shape parameter is known. The asymptotic confidence interval and the highest probability density credible interval are constructed for Rs,k. The reliability estimators are compared by using the estimated risks through Monte Carlo simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, B.C., Balakrishnan, N., Nagaraja, H.N. A First Course in Order Statistics Classics in Applied Mathematics. SIAM, Philadelphia, PA, 2008

    Book  Google Scholar 

  2. Barreto-Souza, W., Lemonte, A.J. Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes. Statistics, 47(6): 1321–1342 (2013)

    Article  MathSciNet  Google Scholar 

  3. Balakrishnan, N., Chan, P.S. Maximum likelihood estimation for the three-parameter log-gamma distribution under Type-II censoring, in Advances in Reliability. CRC Press, Boca Raton, FL, 439–451, 1995

    Google Scholar 

  4. Basirat, M., Baratpour, S., Ahmadi, J. Statistical inferences for stress-strength in the proportional hazard models based on progressive Type-II censored samples. Journal of Statistical Computation and Simulation, 85(3): 431–449 (2015)

    Article  MathSciNet  Google Scholar 

  5. Bhattacharyya, G.K., Johnson, R.A. Estimation of reliability in multicomponent stress-trength model. J. Am. Stat. Assoc., 69: 966–970 (1974)

    Article  Google Scholar 

  6. Chen, M.H., Shao, Q.M. Monte Carlo estimation of Bayesian credible and HPD intervals. J. Comput. Gr. Stat., 8(1): 69–92 (1999)

    MathSciNet  Google Scholar 

  7. Eryilmaz, S. Consecutive k-out-of-n: G system in stress-strength set up. Commun. Stat. Simul. Comput., 37(3): 579–589 (2008)

    Article  Google Scholar 

  8. Eryilmaz, S. Multivariate stress-trength reliability model and its evaluation for coherent structures. J. Multivar. Anal., 99: 1878–1887 (2008)

    Article  Google Scholar 

  9. Eryilmaz, S. On stress-strength reliability with a time-dependent strength. J. Qual. Reliab. Eng., 2013: 1–6 (2013)

    Article  Google Scholar 

  10. Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B. Bayesian data analysis. Chapman Hall, London, 2003

    Book  Google Scholar 

  11. Gradshteyn, I.S., Ryzhik, I.M. Table of integrals, series and products. Academic Press, Boston, 1994

    Google Scholar 

  12. Hanagal, D.D. Estimation of system reliability. Stat. Pap., 40: 99–106 (1999)

    Article  Google Scholar 

  13. KIzIlaslan, F., Nadar, M. Classical and Bayesian estimation of reliability in multicomponent stress-strength model based on Weibull distribution. Rev. Colomb. Estad., 38(2): 467–484 (2015)

    Article  MathSciNet  Google Scholar 

  14. KIzIlaslan, F., Nadar, M. Estimation of reliability in a multicomponent stressCstrength model based on a bivariate Kumaraswamy distribution, Stat. Pape., DOI: https://doi.org/10.1007/s00362-016-0765-8 (2016)

  15. Kuo, W., Zuo, M.J. Optimal reliability modeling, principles and applications. Wiley, New York, 2003

    Google Scholar 

  16. Rao, G.S. Estimation of reliability in multicomponent stress-strength model based on generalized Rayleigh distribution. J. Mod. Appl. Stat. Meth. 13(1): 367–379 (2014)

    Article  Google Scholar 

  17. Rao, G.S., Aslam, M., Kundu, D. Burr Type XII distribution parametric estimation and estimation of reliability of multicomponent stress-strength. Commun. Stat. Theo. Meth., 44(23): 4953–4961 (2015)

    Article  Google Scholar 

  18. Rao, C.R. Linear statistical inference and its applications. Wiley, New York, 1965

    Google Scholar 

  19. Raqab, M.Z. On the maximum likelihood prediction of the exponential distribution based on doubly Type II censored samples, Pakistan J. Stat., 11: 1–10 (1995)

    MathSciNet  Google Scholar 

  20. Sarhan, A.E. Estimation of the mean and standard deviation by order statistics, Part III, Ann. Math. Stat., 26: 576–592 (1955)

    Article  MathSciNet  Google Scholar 

  21. Tierney, L. Markov chains for exploring posterior distributions. Ann. Stat., 22(4): 1701–1728 (1994)

    MathSciNet  Google Scholar 

  22. Wu, S.F. Interval estimation for the two-parameter exponential distribution based on the doubly type II censored sample, Qual. Quan., 41: 489–496 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, 526061, China

    Cong-hua Cheng

Authors
  1. Cong-hua Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cong-hua Cheng.

Ethics declarations

The authors declare no conflict of interest.

Additional information

The project is supported by the Natural Science Foundation of Guangdong (No.2024A1515010983), the project of Guangdong Province General Colleges and Universities with Special Characteristics and Innovations (No.2022KTSCX150), Zhaoqing Science and Technology Innovation Guidance Project (No.2023040317006), Zhaoqing Institute of Education Development Project (No.ZQJYY2023021), Zhaoqing College Quality Project and Teaching Reform Project (No.zlgc202112).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, Ch. Reliability of a Multicomponent Stress-strength Model Based on a Bivariate Kumaraswamy Distribution with Censored Data. Acta Math. Appl. Sin. Engl. Ser. 40, 478–507 (2024). https://doi.org/10.1007/s10255-024-1044-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-024-1044-4

Keywords

2020 MR Subject Classification

Search

Navigation