Published

2015-07-01

Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution

Estimación clásica y bayesiana de la confiabilidad de un modelo estrés-fuerza basado en la distribución Weibull

DOI:

https://doi.org/10.15446/rce.v38n2.51674

Keywords:

Stress-Strength Model, System Reliability, Weibull Distribution (en)
DistribuciónWeibull, Modelo estrés-fuerza, Sistema de confiabilidad. (es)

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Authors

  • Fatih Kizilaslan Gebze Technıcal Unıversıty, Kocaelı, Turkey
  • Mustafa Nadar Istanbul Technıcal Unıversıty, Istanbul, Turkey

In this study, we consider a multicomponent system which has k independent and identical strength components X1,...,Xk and each component is exposed to a common random stress Y when the underlying distributions are Weibull. The system is regarded as operating only if at least s out of k (1 ≤ s ≤ k) strength variables exceeds the random stress. We estimate the reliability of the system by using frequentist and Bayesian approaches. The Bayes estimate of the reliability has been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms. The asymptotic confidence interval and the highest probability density credible interval are constructed for the reliability. The comparison of the reliability estimators is made in terms of the estimated risks by the Monte Carlo simulations.

En este estudio, consideramos un sistema multicomponente con k componentes de fuerza independientes y cada componente expuesto a un estrés común aleatorio Y cuando seconsidera una distribución Bernoulli. El sistema se considera operativo si por lo menos s de los k (1 s k) exceden el estrés aleatorio. Se estima la confiabilidad del sistema usando métodos bayesianos y frecuentistas. La estimación de Bayes de la confiabilidad ha sido desarrollada usando una aproximación de Lindley y métodos MCMC debido a la falta de formas explícitas. El intervalo de confianza asintótico y el intervalo de la densidad deprobabilidad más alta se construyen para la confiabilidad. La comparación de los estimadores de confiabilidad se hace en término de los riesgos estimados por medio de simulaciones Monte Carlo.

https://doi.org/10.15446/rce.v38n2.51674

Classical and Bayesian Estimation of Reliability inMulticomponent Stress-Strength Model Based on Weibull Distribution

Estimación clásica y bayesiana de la confiabilidad deun modelo estrés-fuerza basado en la distribución Weibull

FATIH KIZILASLAN1, MUSTAFA NADAR2

1Gebze Technical University, Faculty of Science, Department of Mathematics, Kocaeli, Turkey. Ph.D. Student. Email: kizilaslan@gtu.edu.tr
2Istanbul Technical University, Faculty of Arts and Sciences, Department of Mathematical Engineering, Istanbul, Turkey. Associate Professor. Email: nadar@itu.edu.tr


Abstract

In this study, we consider a multicomponent system which has k independent and identical strength components X1...,Xk and each component is exposed to a common random stress Y when the underlying distributions are Weibull. The system is regarded as operating only if at least s out of k (1≤ s≤ k) strength variables exceeds the random stress. We estimate the reliability of the system by using frequentist and Bayesian approaches. The Bayes estimate of the reliability has been developed by using Lindleys approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms. The asymptotic confidence interval and the highest probability density credible interval are constructed for the reliability. The comparison of the reliability estimators is made in terms of the estimated risks by the Monte Carlo simulations.

Key words: Stress-Strength Model, System Reliability, Weibull\linebreak Distribution.


Resumen

En este estudio, consideramos un sistema multicomponente con k componentes de fuerzaindependientes y cada componente expuesto a un estrés común aleatorio Y cuando seconsidera una distribución Bernoulli. El sistema se considera operativo si por lo menos s de los k (1≤ s≤ k) exceden el estrés aleatorio. Se estima la confiabilidad del sistema usando métodos bayesianos y frecuentistas. La estimación de Bayes de la confiabilidad ha sido desarrollada usando una aproximación de Lindley y métodos MCMC debido a la falta de formas explícitas. El intervalo de confianza asintótico y el intervalo de la densidad deprobabilidad más alta se construyen para la confiabilidad. La comparación de los estimadores de confiabilidad se hace en término de los riesgos estimados por medio de simulaciones Monte Carlo.

Palabras clave: distribución Weibull, modelo estrés-fuerza, sistema de confiabilidad.


Texto completo disponible en PDF


References

1. Bhattacharyya, G. K. & Johnson, R. A. (1974), 'Estimation of reliability in multicomponent stress-strength model', Journal of the American Statistical Association 69, 966-970.

2. Birnbaum, Z. W. (1956), 'On a use of Mann-Whitney statistics', Proceeding Third Berkeley Symposium on Mathematical Statistics and Probability 1, 13-17.

3. Birnbaum, Z. W. & McCarty, B. C. (1958), 'A distribution-free upper confidence bounds for Pr(Y<X) based on independent samples of X and Y', The Annals of Mathematical Statistics 29(2), 558-562.

4. Chen, M. H. & Shao, Q. M. (1999), 'Monte Carlo estimation of Bayesian credible and HPD intervals', Journal of Computational and Graphical Statistics 8(1), 69-92.

5. Eryilmaz, S. (2008), 'Multivariate stress-strength reliability model and its evaluation for coherent structures', Journal of Multivariate Analysis 99, 1878-1887.

6. Eryilmaz, S. (2010), 'On system reliability in stress-strength setup', Statistics and Probability Letters 80, 834-839.

7. Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. (2003), Bayesian Data Analysis, 2 edn, Chapman & Hall, London.

8. Gradshteyn, I. S. & Ryzhik, I. M. (1994), Table of Integrals, Series and Products, fifth edn, Academic Press, Boston.

9. Hanagal, D. D. (1999), 'Estimation of system reliability', Statistical Papers 40, 99-106.

10. Hanagal, D. D. (2003), 'Estimation of system reliability in multicomponent series stress-strength models', Journal of Indian Statistical Association 41, 1-7.

11. Jae, J. K. & Eun, M. K. (1981), 'Estimation of reliability in a multicomponent stress-strength model in Weibull case', Journal of the Korean Society for Quality Management 9(1), 3-11.

12. Kotz, S., Lumelskii, Y. & Pensky, M. (2003), The Stress-Strength Model and its Generalizations: Theory and Applications, World Scientific, Singapore.

13. Kundu, D. & Gupta, R. D. (2005), 'Estimation of P(Y<X) for generalized exponential distribution', Metrika 61, 291-308.

14. Kundu, D. & Gupta, R. D. (2006), 'Estimation of P(Y<X) for Weibull distribution', IEEE Transactions on Reliability Analysis 52(2), 270-280.

15. Kundu, D. & Raqab, M. Z. (2009), 'Estimation of RP(Y<X) for three-parameter Weibull distribution', Statistics and Probability Letters 79, 1839-1846.

16. Kuo, W. & Zuo, M. J. (2003), Optimal Reliability Modeling, Principles and Applications, John Wiley & Sons, New York.

17. Lindley, D. V. (1980), 'Approximate Bayes method', Trabajos de Estadistica 3, 281-288.

18. Nadar, M., Kizilaslan, F. & Papadopoulos, A. (2014), 'Classical and Bayesian estimation of P(Y <X) for Kumaraswamy's distribution', Journal of Statistical Computation and Simulation 84(7), 1505-1529.

19. Rao, C. R. (1965), Linear Statistical Inference and Its Applications, John Wiley & Sons, New York.

20. Rao, G. S. (2012a), 'Estimation of reliability in multicomponent stress-strength model based on Rayleigh distribution', ProbStat Forum 5, 150-161.

21. Rao, G. S. (2012b), 'Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution', Revista Colombiana de Estadística 35(1), 67-76.

22. Rao, G. S. (2012b), 'Estimation of reliability in multicomponent stress-strength model based on generalized inverted exponential distribution', International Journal of Current Research and Review 4(21), 48-56.

23. Rao, G. S., Aslam, M. & Kundu, D. (2014), 'Burr Type XII distribution parametric estimation and estimation of reliability in multicomponent stress-strength model', Communication in Statistics-Theory and Methods 1.

24. Rao, G. S. & Kantam, R. R. L. (2010), 'Estimation of reliability in multicomponent stress-strength model: Log-logistic distribution', Electronic Journal of Applied Statistical Analysis 3(2), 75-84.

25. Rao, G. S., Kantam, R. R. L., Rosaiah, K. & Reddy, J. P. (2013), 'Estimation of reliability in multicomponent stress-strength model based on inverse Rayleigh distribution', Journal of Statistics Applications & Probability 3, 261-267.

26. Tierney, L. (1994), 'Markov chains for exploring posterior distributions', The Annals of Statistics 22(4), 1701-1728.


[Recibido en junio de 2014. Aceptado en marzo de 2015]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv38n2a10,
    AUTHOR  = {Kizilaslan, Fatih and Nadar, Mustafa},
    TITLE   = {{Classical and Bayesian Estimation of Reliability inMulticomponent Stress-Strength Model Based on Weibull Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {2},
    pages   = {467-484}
}

References

Bhattacharyya, G. K. & Johnson, R. A. (1974), ‘Estimation of reliability in multicomponent stress-strength model’, Journal of the American Statistical Association 69, 966–970.

Birnbaum, Z. W. (1956), ‘On a use of Mann-Whitney statistics’, Proceeding Third Berkeley Symposium on Mathematical Statistics and Probability 1, 13–17.

Birnbaum, Z. W. & McCarty, B. C. (1958), ‘A distribution-free upper confidence bounds for Pr(Y < X) based on independent samples of X and Y ’, The Annals of Mathematical Statistics 29(2), 558–562.

Chen, M. H. & Shao, Q. M. (1999), ‘Monte Carlo estimation of Bayesian credible and HPD intervals’, Journal of Computational and Graphical Statistics 8(1), 69–92.

Eryilmaz, S. (2008), ‘Multivariate stress-strength reliability model and its evaluation for coherent structures’, Journal of Multivariate Analysis 99, 1878–1887.

Eryilmaz, S. (2010), ‘On system reliability in stress-strength setup’, Statistics and Probability Letters 80, 834–839.

Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. (2003), Bayesian Data Analysis, 2 edn, Chapman & Hall, London.

Gradshteyn, I. S. & Ryzhik, I. M. (1994), Table of Integrals, Series and Products, fifth edn, Academic Press, Boston.

Hanagal, D. D. (1999), ‘Estimation of system reliability’, Statistical Papers 40, 99–106.

Hanagal, D. D. (2003), ‘Estimation of system reliability in multicomponent series stress-strength models’, Journal of Indian Statistical Association 41, 1–7.

Jae, J. K. & Eun, M. K. (1981), ‘Estimation of reliability in a multicomponent stress-strength model in Weibull case’, Journal of the Korean Society for Quality Management 9(1), 3–11.

Kotz, S., Lumelskii, Y. & Pensky, M. (2003), The Stress-Strength Model and its Generalizations: Theory and Applications, World Scientific, Singapore.

Kundu, D. & Gupta, R. D. (2005), ‘Estimation of P(Y < X) for generalized exponential distribution’, Metrika 61, 291–308.

Kundu, D. & Gupta, R. D. (2006), ‘Estimation of P(Y < X) for Weibull distribution’, IEEE Transactions on Reliability Analysis 52(2), 270–280.

Kundu, D. & Raqab, M. Z. (2009), ‘Estimation of R = P(Y < X) for threeparameter Weibull distribution’, Statistics and Probability Letters 79, 1839–1846.

Kuo, W. & Zuo, M. J. (2003), Optimal Reliability Modeling, Principles and Applications, John Wiley & Sons, New York.

Lindley, D. V. (1980), ‘Approximate Bayes method’, Trabajos de Estadistica 3, 281–288.

Nadar, M., Kizilaslan, F. & Papadopoulos, A. (2014), ‘Classical and Bayesian estimation of P(Y < X) for Kumaraswamy’s distribution’, Journal of Statistical Computation and Simulation 84(7), 1505–1529.

Rao, C. R. (1965), Linear Statistical Inference and Its Applications, John Wiley & Sons, New York.

Rao, G. S. (2012a), ‘Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution’, Revista Colombiana de Estadística 35(1), 67–76.

Rao, G. S. (2012b), ‘Estimation of reliability in multicomponent stress-strength model based on generalized inverted exponential distribution’, International Journal of Current Research and Review 4(21), 48–56.

Rao, G. S. (2012c), ‘Estimation of reliability in multicomponent stress-strength model based on Rayleigh distribution’, ProbStat Forum 5, 150–161.

Rao, G. S., Aslam, M. & Kundu, D. (2014), ‘Burr Type XII distribution parametric estimation and estimation of reliability in multicomponent stress-strength model’, Communication in Statistics-Theory and Methods 1.

Rao, G. S. & Kantam, R. R. L. (2010), ‘Estimation of reliability in multicomponent stress-strength model: log-logistic distribution’, Electronic Journal of Applied Statistical Analysis 3(2), 75–84.

Rao, G. S., Kantam, R. R. L., Rosaiah, K. & Reddy, J. P. (2013), ‘Estimation of reliability in multicomponent stress-strength model based on inverse Rayleigh distribution’, Journal of Statistics Applications & Probability 3, 261–267.

Tierney, L. (1994), ‘Markov chains for exploring posterior distributions’, The Annals of Statistics 22(4), 1701–1728.

How to Cite

APA

Kizilaslan, F. and Nadar, M. (2015). Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution. Revista Colombiana de Estadística, 38(2), 467–484. https://doi.org/10.15446/rce.v38n2.51674

ACM

[1]
Kizilaslan, F. and Nadar, M. 2015. Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution. Revista Colombiana de Estadística. 38, 2 (Jul. 2015), 467–484. DOI:https://doi.org/10.15446/rce.v38n2.51674.

ACS

(1)
Kizilaslan, F.; Nadar, M. Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution. Rev. colomb. estad. 2015, 38, 467-484.

ABNT

KIZILASLAN, F.; NADAR, M. Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution. Revista Colombiana de Estadística, [S. l.], v. 38, n. 2, p. 467–484, 2015. DOI: 10.15446/rce.v38n2.51674. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/51674. Acesso em: 28 mar. 2024.

Chicago

Kizilaslan, Fatih, and Mustafa Nadar. 2015. “Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution”. Revista Colombiana De Estadística 38 (2):467-84. https://doi.org/10.15446/rce.v38n2.51674.

Harvard

Kizilaslan, F. and Nadar, M. (2015) “Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution”, Revista Colombiana de Estadística, 38(2), pp. 467–484. doi: 10.15446/rce.v38n2.51674.

IEEE

[1]
F. Kizilaslan and M. Nadar, “Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution”, Rev. colomb. estad., vol. 38, no. 2, pp. 467–484, Jul. 2015.

MLA

Kizilaslan, F., and M. Nadar. “Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution”. Revista Colombiana de Estadística, vol. 38, no. 2, July 2015, pp. 467-84, doi:10.15446/rce.v38n2.51674.

Turabian

Kizilaslan, Fatih, and Mustafa Nadar. “Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution”. Revista Colombiana de Estadística 38, no. 2 (July 1, 2015): 467–484. Accessed March 28, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/51674.

Vancouver

1.
Kizilaslan F, Nadar M. Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution. Rev. colomb. estad. [Internet]. 2015 Jul. 1 [cited 2024 Mar. 28];38(2):467-84. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/51674

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