Published

2017-01-16

Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution

En estimación del estrés fuerza modelo en la caja P(Yr:n1 < Xk:n2 ) de distribución Lindley

DOI:

https://doi.org/10.15446/rce.v40n1.54349

Keywords:

Bayes Estimator, Lindley Distribution, Maximum Likelihood Estimator, Order Statistics, Stress-Strength Model, Uniformly Minimum Variance Unbiased Estimator (en)
Distribución de Lindley, estadísticas de orden, estimador de Bayes, estimador insesgado de varianza uniformemente mínima, estimador insesgado de varianza mínima, modelo de estrés-fuerza (es)

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Authors

  • Marwa Khalil Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

The problem of estimation reliability in a multicomponent stress-strength model, when the system consists of k components have strength each compo- nent experiencing a
random stress, is considered in this paper. The reliability of such a system is obtained when strength and stress variables are given by Lindley distribution. The system is regarded as alive only if at least r out of k (r < k) strength exceeds the stress. The multicomponent reliability of the system is given by Rr,k . The maximum likelihood estimator (M LE), uniformly minimum variance unbiased estimator (UMVUE) and Bayes esti- mator of Rr,k are obtained. A simulation study is performed to compare the different estimators of Rr,k . Real data is used as a practical application of the proposed model.

El problema de la fiabilidad de estimación en el modelo de estrés-fuerza multicomponente, cuando el sistema consta de componentes k tiene fuerza, cada componente experimentando un estrés al azar se considera en este documento. Se obtiene la fiabilidad de estos sistemas cuando las variables de fuerza y tensión están dadas por la distribución Lindley. El sistema es considerado como vivo solo si al menos r de k(r < k) fuerzas superan el estrés. La fiabilidad de varios componentes del sistema viene dado por Rr;k. El estimador de máxima verosimilitud (MLE), se obtienen estimadores insesgados de varianza uniformemente mínima (UMVUE) y el estimador de Bayes Rr;k. Se realizó un estudio de simulación para comparar los diferentes estimadores de Rr;k. Se utilizaron datos reales como aplicación práctica para el modelo propuesto.

https://doi.org/10.15446/rce.v40n1.54349

Estimation a Stress-Strength Model for P(Yr:n_1< Xk:n_2) Using the Lindley Distribution

En estimación del estrés fuerza modelo en la caja P(Yr:n_1< Xk:n_2) de distribución Lindley

MARWA KH HASSAN1

1Ain Shams University, Faculty of Education, Department of Mathematics, Cairo, Egypt. PhD. Email: marwa_khalil2006@hotmail.com

Abstract

The problem of estimation reliability in a multicomponent stress-strength model, when the system consists of k components have strength each component experiencing a random stress, is considered in this paper. The reliability of such a system is obtained when strength and stress variables are given by Lindley distribution. The system is regarded as alive only if at least r out of k (r< k) strength exceeds the stress. The multicomponent reliability of the system is given by Rr,k. The maximum likelihood estimator (MLE), uniformly minimum variance unbiased estimator (UMVUE) and Bayes estimator of Rr,k are obtained. A simulation study is performed to compare the different estimators of Rr,k. Real data is used as a practical application of the proposed model.

Key words: Bayes Estimator, Lindley Distribution, Maximum Likelihood Estimator, Order Statistics, Stress-Strength Model, Uniformly Minimum Variance Unbiased Estimator.

Resumen

El problema de la fiabilidad de estimación en el modelo de estrés-fuerza multicomponente, cuando el sistema consta de componentes k tiene fuerza, cada componente experimentando un estrés al azar se considera en este documento. Se obtiene la fiabilidad de estos sistemas cuando las variables de fuerza y tensión están dadas por la distribución Lindley. El sistema es considerado como vivo solo si al menos r de k (r < k) fuerzas superan el estrés. La fiabilidad de varios componentes del sistema viene dado por Rr, k . El estimador de máxima verosimilitud (MLE), se obtienen estimadores insesgados de varianza uniformemente mínima (UMVUE) y el estimador de Bayes Rr, k . Se realizó un estudio de simulación para comparar los diferentes estimadores de Rr, k . Se utilizaron datos reales como aplicación práctica para el modelo propuesto.

Palabras clave: distribución de Lindley, estadísticas de orden, estimador de Bayes, estimador insesgado de varianza uniformemente mínima, estimador insesgado de varianza mínima, modelo de estrés-fuerza.

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References

1. Al-Mutairi, D. K., Ghitany, M. E. & Kundu, D. (2013), 'Inferences on stress-strength reliability from lindley distributions', Communications in Statistics Theory and Methods 42(8), 1443-1463.

2. Ali, M., Pal, M. & Woo, J. (2012), 'Estimation of P(Y < X) in a four-parameter generalized Gamma distribution', Austrian Journal of Statistics 41(3), 197-210.

3. Basu, A. P. (1964), 'Estimates of reliability for some distributions useful in reliability', Technometrics 6, 215-219.

4. Beg, M. A. (1980), 'On the estimation of P(Y < X) for two-parameter exponential distribution', Metrika 80, 29-34.

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

6. Birnbaum, Z. W. (1956), On a use of the mann-whitney statistic, 'Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability', Vol. 1, University of California Press, Berkeley, CA., p. 13-17.

7. Dey, S., Mazucheli, J. & Anis, M. Z. (2017), 'Estimation of reliability of multicomopnent stress-strength for kumaraswamy distribution', Communications in Statistics - Theory and Methods 46(4), 1560-1572.

8. Downtown, F. (1973), 'The estimation of P(Y < X) in the normal case', Technometrics 15, 551-558.

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

10. Ghitany, M. E., Al-Mutairi, D. K. & Aboukhamseen, S. M. (2015), 'Estimation of the reliability of a stress-strength system from power Lindley distributions', Communications in Statistics Simulation and Computation 44, 118-136.

11. Ghitany, M. E. & Atieh, B. a. N. S. (2008), 'Lindley distribution and it's application', Journal of Mathematics and Computers in Simulation 76, 493-506.

12. Hussian, M. A. (2013), 'On estimation of stress-strength model for generalized inverted Exponential distribution', Journal of Reliability and Statistical Studies 6(2), 55-63.

13. Iwase, K. (1987), 'On UMVU estimators of Pr (Y< X) in the 2-parameter exponential case', Memoirs of the Faculty of Engineering, Hiroshima University 9, 21-24.

14. Jeffrey, H. (1961), Theory of probability, 3 edn, Oxford University Press.

15. Kizilaslan, F. & 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.

16. Krishna, H. & Kumar, K. (2011), 'Reliability estimation in Lindley distribution with progressively type-II right censored sample', Mathematics and Computers in Simulation 82, 281-294.

17. Lawless, J. F. (1982), Statistical Models and Methods for Lifetime Data, John Wiley & Sons, Inc.

18. Lindley, D. V. (1958), 'Fiducial distributions and bayes' theorem', Journal of the Royal Statistical Society. Series B (Methodological), 102-107.

19. Lindley, D. V. (1965), Introduction to Probability and Statistics from a Bayesian Viewpoint, Vol. 2, Combridge University Press, New York.

20. Lindley, D. V. (1980), 'Approximate bayesian methods', Trabajos de estadística y de investigación operativa 31(1), 223-245.

21. McCool, J. I. (1991), 'Inference on P(X < Y ) in the Weibull case', Communications in Statistcs Simulation and Computation 20, 129-148.

22. Najarzadegan, H., Babaii, S., Rezaei, S. & Nadarajah, S. (2016), 'Estimation of P(Y < X) for the Levy distribution', Hacettepe Bulletin of Natural Sciences and Engineering 45(3), 957-972.

23. Pakdaman, Z. & Ahmadi, J. (2013), 'Stress-strength reliability for P[X_r:n_1,k:n_2] in exponential case', Journal of The Turkish Statistical Association 6(3), 92-102.

24. Pandey, M., Uddin, M. B. & Ferdous, J. (1992), 'Reliability estimation of AN s-out-of-k system with non-identical component strengths: the Weibull case', Reliability Engineering & System Safety 36(2), 109-116.

25. Proschan, F. (1963), 'Theoretical explanation of observed decreasing failure rate', Technometrics 5, 375-383.

26. Rao, C. R. (1973), Linear statistical inference and application, Jon Wiley and Sons, New York.

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

28. Rao, G. S. & Kantan, 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.

29. Rao, G. S., Muhammad, A. & Osama, H. (2016), 'Estimation of reliability in multicomponent stress-strength based on two parameter exponentiated Weibull distribution', Communications in Statistics-Theory and Methods(10.1080/03610926.2016.1154155).

30. Shahsanaei, F. & Daneshkhah, A. (2013), 'Estimation of stress-strength model in the generalized linear failure rate distribution', arXiv preprint arXiv:1312.0401.

31. Sharma, V. K., Singh, S. K., Singh, U. & Agiwal, V. (2014), 'The inverse Lindley distribution: a stress-strength reliability model', arXiv preprint arXiv:1405.6268.

32. Sharma, V. K., Singh, S. K., Singh, U. & Agiwal, V. (2015), 'The inverse lindley distribution: a stress-strength reliability model with application to head and neck cancer data', Journal of Industrial and Production Engineering 32(3), 162-173.

33. Singh, B., Gupta, P. K. & Sharma, V. K. (2014), 'On type-ii hybrid censored lindley distribution', Statistics Research Letters 3, 58-62.

34. Singh, P. K., Singh, S. K. & Singh, U. (2008), 'Bayes estimator of inverse gaussian parameters under general entropy loss function using lindley's approximation', Communications in Statistics-Simulation and Computation 37(9), 1750-1762.

35. Tong, H. (1974), 'A note on the estimation of P(Y < X) in the Exponential case', Technometrics 16(4), 625-625.

36. Tong, H. (1977), 'On the estimation of P(Y < X) for Exponential families', IEEE Transactions on Reliability 1, 54-56.

37. Wong, A. (2012), 'Interval estimation of P (Y< X) for generalized Pareto distribution', Journal of Statistical Planning and Inference 142(2), 601-607.

[Recibido en noviembre de 2015. Aceptado en noviembre de 2016]

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

@ARTICLE{RCEv40n1a05,
    AUTHOR  = {Hassan, Marwa KH},
    TITLE   = {{Estimation a Stress-Strength Model for P( Yr:n_1< Xk:n_2 ) Using the Lindley Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2017},
    volume  = {40},
    number  = {1},
    pages   = {105-121}
}

References

Al-Mutairi, D. K., Ghitany, M. E. & Kundu, D. (2013), ‘Inferences on stressstrength

reliability from lindley distributions’, Communications in Statistics Theory and Methods 42(8), 1443–1463.

Ali, M., Pal, M. & Woo, J. (2012), ‘Estimation of p(y < x) in a four-parameter generalized Gamma distribution’, Austrian Journal of Statistics 41(3), 197–210.

Basu, A. P. (1964), ‘Estimates of reliability for some distributions useful in reliability’,

Technometrics 6, 215–219.

Beg, M. A. (1980), ‘On the estimation of p(y < x) for two-parameter exponential distribution’, Metrika 80, 29–34.

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

Birnbaum, Z. W. (1956), On a use of the mann-whitney statistic, in ‘Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability’, Vol. 1, University of California Press, Berkeley, CA., pp. 13–17.

Dey, S., Mazucheli, J. & Anis, M. Z. (2017), ‘Estimation of reliability of multicomopnent

stress-strength for kumaraswamy distribution’, Communications in Statistics - Theory and Methods 46(4), 1560–1572.

Downtown, F. (1973), ‘The estimation of p(y < x) in the normal case’, Technometrics

, 551–558.

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

Ghitany, M. E., Al-Mutairi, D. K. & Aboukhamseen, S. M. (2015), ‘Estimation of the reliability of a stress-strength system from power Lindley distributions’, Communications in Statistics Simulation and Computation 44, 118–136.

Ghitany, M. E. & Atieh, B. ans Nadarajah, S. (2008), ‘Lindley distribution and it’s application’, Journal of Mathematics and Computers in Simulation 76, 493– 506.

Hussian, M. A. (2013), ‘On estimation of stress-strength model for generalized inverted Exponential distribution’, Journal of Reliability and Statistical Studies 6(2), 55–63.

Iwase, K. (1987), ‘On UMVU estimators of Pr(Y < X) in the 2-parameter exponential case’, Memoirs of the Faculty of Engineering, Hiroshima University 9, 21–24.

Jeffrey, H. (1961), Theory of probability, 3 edn, Oxford University Press.

Kizilaslan, F. & 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.

Krishna, H. & Kumar, K. (2011), ‘Reliability estimation in Lindley distribution with progressively type-II right censored sample’, Mathematics and Computers in Simulation 82, 281–294.

Lawless, J. F. (1982), Statistical Models and Methods for Lifetime Data, John Wiley & Sons, Inc.

Lindley, D. V. (1958), ‘Fiducial distributions and bayes’ theorem’, Journal of the Royal Statistical Society. Series B (Methodological) pp. 102–107.

Lindley, D. V. (1965), Introduction to Probability and Statistics from a Bayesian Viewpoint, Vol. 2, Combridge University Press, New York.

Lindley, D. V. (1980), ‘Approximate bayesian methods’, Trabajos de estadística y de investigación operativa 31(1), 223–245.

McCool, J. I. (1991), ‘Inference on p(x < y) in the Weibull case’, Communications in Statistcs Simulation and Computation 20, 129–148.

Najarzadegan, H., Babaii, S., Rezaei, S. & Nadarajah, S. (2016), ‘Estimation of p(y < x) for the Levy distribution’, Hacettepe Bulletin of Natural Sciences and Engineering 45(3), 957–972.

Pakdaman, Z. & Ahmadi, J. (2013), ‘Stress-strength reliability for p[xr:n1;k:n2 ] in exponential case’, Journal of The Turkish Statistical Association 6(3), 92–102.

Pandey, M., Uddin, M. B. & Ferdous, J. (1992), ‘Reliability estimation of AN s-out-of-k system with non-identical component strengths: the Weibull case’, Reliability Engineering & System Safety 36(2), 109–116.

Proschan, F. (1963), ‘Theoretical explanation of observed decreasing failure rate’, Technometrics 5, 375–383.

Rao, C. R. (1973), Linear statistical inference and application, Jon Wiley and Sons, New York.

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

Rao, G. S. & Kantan, 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., Muhammad, A. & Osama, H. (2016), ‘Estimation of reliability in multicomponent stress-strength based on two parameter exponentiated Weibull distribution’, Communications in Statistics-Theory and Methods (10.1080/03610926.2016.1154155).

Shahsanaei, F. & Daneshkhah, A. (2013), ‘Estimation of stress-strength model in the generalized linear failure rate distribution’, arXiv preprint arXiv:1312.0401.

Sharma, V. K., Singh, S. K., Singh, U. & Agiwal, V. (2014), ‘The inverse Lindley distribution: A stress-strength reliability model’, arXiv preprint arXiv:1405.6268.

Sharma, V. K., Singh, S. K., Singh, U. & Agiwal, V. (2015), ‘The inverse lindley distribution: a stress-strength reliability model with application to head and neck cancer data’, Journal of Industrial and Production Engineering 32(3), 162–173.

Singh, B., Gupta, P. K. & Sharma, V. K. (2014), ‘On type-ii hybrid censored lindley distribution’, Statistics Research Letters 3, 58–62.

Singh, P. K., Singh, S. K. & Singh, U. (2008), ‘Bayes estimator of inverse Gaussian parameters under general entropy loss function using lindley’s approximation’, Communications in Statistics-Simulation and Computation 37(9), 1750–1762.

Tong, H. (1974), ‘A note on the estimation of p(y < x) in the Exponential case’, Technometrics 16(4), 625–625.

Tong, H. (1977), ‘On the estimation of p(y < x) for Exponential families’, IEEE Transactions on Reliability 1, 54–56.

Wong, A. (2012), ‘Interval estimation of p(y < x) for generalized Pareto distribution’, Journal of Statistical Planning and Inference 142(2), 601–607.

How to Cite

APA

Khalil, M. (2017). Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution. Revista Colombiana de Estadística, 40(1), 105–121. https://doi.org/10.15446/rce.v40n1.54349

ACM

[1]
Khalil, M. 2017. Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution. Revista Colombiana de Estadística. 40, 1 (Jan. 2017), 105–121. DOI:https://doi.org/10.15446/rce.v40n1.54349.

ACS

(1)
Khalil, M. Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution. Rev. colomb. estad. 2017, 40, 105-121.

ABNT

KHALIL, M. Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution. Revista Colombiana de Estadística, [S. l.], v. 40, n. 1, p. 105–121, 2017. DOI: 10.15446/rce.v40n1.54349. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/54349. Acesso em: 28 mar. 2025.

Chicago

Khalil, Marwa. 2017. “Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution”. Revista Colombiana De Estadística 40 (1):105-21. https://doi.org/10.15446/rce.v40n1.54349.

Harvard

Khalil, M. (2017) “Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution”, Revista Colombiana de Estadística, 40(1), pp. 105–121. doi: 10.15446/rce.v40n1.54349.

IEEE

[1]
M. Khalil, “Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution”, Rev. colomb. estad., vol. 40, no. 1, pp. 105–121, Jan. 2017.

MLA

Khalil, M. “Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution”. Revista Colombiana de Estadística, vol. 40, no. 1, Jan. 2017, pp. 105-21, doi:10.15446/rce.v40n1.54349.

Turabian

Khalil, Marwa. “Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution”. Revista Colombiana de Estadística 40, no. 1 (January 1, 2017): 105–121. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/54349.

Vancouver

1.
Khalil M. Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution. Rev. colomb. estad. [Internet]. 2017 Jan. 1 [cited 2025 Mar. 28];40(1):105-21. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/54349

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CrossRef Cited-by

CrossRef citations6

1. Elisavet Demertzi, Stelios Psarakis. (2024). Probability‐type, Shewhart‐type and EWMA control charts for individual observations from the two‐parameter Lindley distribution. Quality and Reliability Engineering International, 40(1), p.313. https://doi.org/10.1002/qre.3401.

2. Abbas Pak, Arjun Kumar Gupta, Nayereh Bagheri Khoolenjani. (2018). On Reliability in a Multicomponent Stress-Strength Model with Power Lindley Distribution. Revista Colombiana de Estadística, 41(2), p.251. https://doi.org/10.15446/rce.v41n2.69621.

3. Prashant Kumar Sonker, Mukesh Kumar, Agni Saroj. (2023). Stress–strength reliability models on power-Muth distribution. International Journal of System Assurance Engineering and Management, 14(S1), p.173. https://doi.org/10.1007/s13198-022-01832-w.

4. Marwa KH. Hassan. (2022). On estimating of the stress-strength model with more than two states under Gompertz distribution. Journal of Statistics and Management Systems, 25(3), p.665. https://doi.org/10.1080/09720510.2021.1947454.

5. Marwa K. H. Hassan, Abdisalam Hassan Muse, Sania Qureshi. (2023). Fuzzy Stress‐Strength Model and Mean Remaining Strength for Lindley Distribution: Estimation and Application in Cancer of Benign Endocrine. Computational and Mathematical Methods in Medicine, 2023(1) https://doi.org/10.1155/2023/8952946.

6. Fatih Kızılaslan. (2019). The mean remaining strength of parallel systems in a stress-strength model based on exponential distribution. Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 68(2), p.1435. https://doi.org/10.31801/cfsuasmas.539171.

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