Using two Reliability Estimators in a Stress-Strength System

Subject Areas : Statistics

Kazem Fayyaz Heidari ¹ , Einolah Deiri ² , Ezzatallah baloui Jamkhaneh ³

1 - Department of statistic, Islamic Azad University
2 - Department of statistic, Islamic Azad University
3 - Department of statistic, Islamic Azad University

Received: 2020-12-06 Accepted : 2021-05-27 Published : 2022-08-23

Keywords: توزیع گومپرتز, بهترین برآوردگر صدکی, استرس-مقاومت چند مولفه ای, قابلیت اطمینان,

Abstract :

In this paper, we propose an estimate of reliability in a multicomponent stress-strength system. The reliability of such a system is obtained when strength and stress variables are given by Gompertz distribution with common scale parameter λ and different shape parameters α and . The system reliability is estimated using maximum likelihood estimation (MLE) and the best observational percentile estimation (BPE) methods in samples drawn from strength and stress distributions. Also, the asymptotic confidence interval for system reliability is obtained. The reliability estimators obtained from both methods are compared using mean squares error criteria and confidence interval length via Monte Carlo simulation. In the end, using two real data sets we illustrate the procedure. Before analyzing the data, we first show that the Gompertz distribution is fitted to this data sets using the Kolmogorov-Smirnov goodness-of-fit test statistic. In general, the simulation results show that due to the increase in sample size, the average mean, mean squares error and length of confidence interval in the maximum likelihood method is decreasing compared to the best percentile estimator method. This indicates that the maximum likelihood method is more efficient. Also, in this paper, it was shown that the one out of three component system reliability is more than the one out of two component system reliability for both methods of estimation.

References:

1] Gompertz, B. (1825). On the Nature of the Function Expressive of the Law of Human Mortality and on the New Mode of Determining the Value of Life ‎Contingencies, ‎Philosophical Transactions of the Royal Society American, 115, 513-580.

 

[2] Bhattacharyya,G.K. and Johnson, R.A. (1974). Estimation of reliability in multicomponent stress-strengthmodel,JASA, 69, 966–970.

 

[3] Saraçoglu, B., Kaya, M. F and Abd-Elfattah, A. M. (2009). Comparison of estimators forstress-strength reliability in the Gompertz case, Hacettepe. Math Statist, 38, 339-349.

 

[4] Al-Mutairi, D. K., Gitany, M. E. and Kundu, D. (2013). Inferences on Stress-strength reliability from Lindley distribution, Communications in Statistics-Theory and Methods, 42(8), 1443-1463.

 

[5] Asgharzadeh, A., Valiollahi, R. and Raqab, M. Z. (2011). Stress-strength reliability of Weibull distribution based on progressively censord samples, SORT, 35(2), 103-124.

 

[6] Awad, M. and Gharraf, K. (1986). Estimation of P (Y < X) in Burr case: A comparative study, Communications in Statistics - Simulation and Computation, 15, 389-403.

 

[7] Davarzani, N. Haghighi, F and Parsian, A. (2009).  Estimation of P(X < Y) for a Bivariate Weibull distribution, Appl. Probability Statist, 4, 227-238.

 

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

 

[9] Nadar, M, Kızılaslan. F and Papadopoulos, A. (2014). Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution, Statist. Computat. Simulat., 84, 1505-1529.

 

[10] Raqab, M.Z. and Kundu, D. (2005). Comparison of di_erent estimators of P (Y < X) for a scaled Burr type X distribution, Communications in Statistics - Simulation and Computation, 34(2), 465-483.

 

[11] Pandey, M. and Borhan U, M. d. (1985). Estimation of reliability in multicomponent stress-strength model following Burr distribution, Proceedings of the First Asian congress on Quality and Reliability, New Delhi, India, 307-312.

[12] Heidari, K.F., Deiri, E. & Jamkhaneh, E.B (2021). Using the best two-observational percentile and maximum likelihood methods in a multicomponent stress-strength system to reliability estimation of inverse Weibull distribution. Life Cycle Reliab Saf Eng. https://doi.org/10.1007/s41872-021-00166-z.

[13] Eryilmaz, S and Iscioglu, F. (2011). Reliability evaluation for a multi-state system under stress-strength setup, Commun. Statist. Theory Methods, 40, 547-558.

 

[14] Pak, A, Khoolenjani. N. B and Khorshidian, K.(2009). Estimation of system reliability under bivariate Rayleigh distribution, Econ. Quality Control, 24, 143-151.

 

[15] Rao, G.S. (2012). Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution, Colombian Journal of Statistics,35(1), 67-76.

 

[16] Rao, G.S., Aslam, M. and Aril, O. H. (2017). Estimation of reliability in multicomponent stress-strength model based on exponentiated Weibull distribution, Communications Statistics Simulation and Computation, 46(15), 7495-7502.

 

[17] Rao, G.S. and 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.

 

[18] Travadi, R. J. and Ratani, R.T. (1990). On estimation of reliability function for inverse Guassian distribution with known coe_cientof variation, IAPQR Transactions,5, 29-37.

 

[19] Kantam, R.R.L. and Rao G.S. (2002). Log-logistic distribution: Modi_ed Maximum likelihood estimation, Gujarat Statistical Review,29(1-2), 25-36.

 

[20] Rao, G.S. (2012). Estimation of reliability in multicomponent stress-strength model based on Rayliegh distribution, ProbStat Forun, 5, 150-161.