Isaac Scientific Publishing

Journal of Advanced Statistics

Estimation of the Stress-Strength Reliability for the Dagum Distribution

Download PDF (523.7 KB) PP. 156 - 170 Pub. Date: September 1, 2016

DOI: 10.22606/jas.2016.13005

Author(s)

  • Bander Al-Zahrani*
    Department of Statistics, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia
  • Samerah Basloom
    Department of Statistics, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia

Abstract

The paper deals with the problem of estimating the reliability of a single and multicomponent stress-strength based on a three-parameter Dagum distribution. The maximum likelihood estimators and the Bayes estimators of R = P(Y < X) when X and Y are two independent random variables follow Dagum distribution and their asymptotic distributions are obtained. Also, the reliability of a multicomponent stress-strength model is estimated using the maximum likelihood. Consequently, the asymptotic confidence intervals of the reliability of a single and multicomponent stress-strength model are constructed. Gibbs and Metropolis sampling were used to provide sample-based estimates of the reliability and its associated credible intervals. Finally, Monte Carlo simulations are carried out for illustrative purposes.

Keywords

Dagum distribution, maximum likelihood estimator, Bayes estimator, stress-strength models.

References

[1] C. Dagum, A New Model for Personal Income Distribution: Specification and Estimation, Economic Applique, 33 (1977) 413–437.

[2] F. Domma, L’andamento della Hazard function nel modello di Dagum a tre parametri, Quaderni di Statistica, 4 (2002), 103–114.

[3] F. Domma S. Giordano and M. M. Zenga, Maximum likelihood estimation in Dagum distribution with censored sample, Journal of Applied Statistics, 38 (2011), 2971–2985.

[4] F. Domma and F. Condino, The beta-Dagum distribution: definition and properties, Communications in Statistics-Theory and Methods, 42 (2013) 4070–4090.

[5] C. Kleiber and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences. (John Wiley, Hoboken, New Jersey, 2003).

[6] C. Kleiber, A guide to the Dagum distributions. (Springer, New York, 2008).

[7] M. N. Shahzad and Z. Asghar, Comparing TL-Moments, L-Moments and Conventional Moments of Dagum Distribution by Simulated data, Revista Colombiana de Estadistica 36 (2013) 79–93.

[8] B. O. Oluyede and Y. Ye, Weighted Dagum and related distributions, Afrika Matematika, 25 (2014) 1125–1141.

[9] S. Kotz, Y. Lumelskii, and M. Pensky, The Stress-Strength Model and its Generalizations-Theory and Applications. (World Scientific, New York, 2003)

[10] M. Pandey and B. Uddin, . Estimation of reliability in multicomponent stress-strength model following Burr distribution. (Proceedings of the First Asian congress on Quality and Reliability, (pp. 307–312). New Delhi, India 1985)

[11] R. G. Srinivasa and R. R. L. Kantam, Estimation of reliability in multicomponent stress-strength model: log-logistic distribution, Electronic Journal of Applied Statistical Analysis, 3 (2010) 75–84.

[12] S. Nadarajah and S. Kotz, Moments of some J-shaped distributions, Journal of Applied Statistics, 30 (2003) 311–317.

[13] G. Casella and B. L. Roger, Statistical inference. (Vol. 2. Pacific Grove, CA: Duxbury, 2002)

[14] A. Asgharzadeh, R. Valiollahi and M. Z. Raqab, Estimation of the stress-strength reliability for the generalized logistic distribution, Statistical Methodology 15 (2013) 73–94.

[15] L. Held and D. S Bove, Applied Statistical Inference: Likelihood and Bayes. (Springer Science & Business Media, 2013)

[16] J. F. Lawless, Statistical Models and Methods for Lifetime Data, (Second ed., John Wiley and Sons, New York, 2003).

[17] S. P. Mukherjee and S. S. Maiti, Stress-strength reliability in the Weibull case, Frontiers in Reliability 4 (1998) 231–248.

[18] P. Hall, Theoretical comparison of bootstrap confidence intervals, Annals of Statistics 16 (1988) 927–953.

[19] M. H. Chen, Q. M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, Journal of Computational and Graphical Statistics 8 (1999) 69–92.

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

[21] C. R. Rao, Linear Statistical Inference and its Applications, (Wiley Eastern Limited, India 1973)