zbMATH — the first resource for mathematics

Estimation of the stress-strength reliability for the generalized logistic distribution. (English) Zbl 07035617
Summary: A. Ragab [“Estimation and predictive density for the generalized logistic distribution”, Microelectron. Reliab. 31, No. 1, 91–95 (1991; doi:10.1016/0026-2714(91)90353-9)] described the Bayesian and empirical Bayesian methods for estimation of the stress-strength parameter \(R = P(Y < X)\), when \(X\) and \(Y\) are independent random variables from two generalized logistic (GL) distributions having the same known scale but different shape parameters. In this current paper, we consider the estimation of \(R\), when \(X\) and \(Y\) are both two-parameter GL distribution with the same unknown scale but different shape parameters or with the same unknown shape but different scale parameters. We also consider the general case when the shape and scale parameters are different. The maximum likelihood estimator of \(R\) and its asymptotic distribution are obtained and it is used to construct the asymptotic confidence interval of \(R\). We also implement Gibbs and Metropolis samplings to provide a sample-based estimate of \(R\) and its associated credible interval. Finally, analyzes of real data set and Monte Carlo simulation are presented for illustrative purposes.

62-XX Statistics
PDF BibTeX Cite
Full Text: DOI
[1] Ahmad, K. E.; Fakhry, M. E.; Jaheen, Z. F., Empirical Bayes estimation of \(P(Y < X)\) and characterizations of the burr-type \(X\) model, Journal of Statistical Planning and Inference, 64, 297-308, (1997) · Zbl 0915.62001
[2] Alkasasbeh, M. R.; Raqab, M. Z., Estimation of the generalized logistic distribution parameters: comparative study, Statistical Methodology, 6, 262-279, (2009) · Zbl 1463.62045
[3] Asgharzadeh, A., Point and interval estimation for a generalized logistic distribution under progressive type-II censoring, Communications in Statistics—Theory and Methods, 35, 1685-1702, (2006) · Zbl 1105.62093
[4] Asgharzadeh, A.; Valiollahi, R.; Raqab, M. Z., Stress-strength reliability of Weibull distribution based on progressively censored samples, SORT. Statistics and Operations Research Transactions, 35, 2, 103-124, (2011) · Zbl 1284.62144
[5] Awad, A. M.; Azzam, M. M.; Hamadan, M. A., Some inference results in \(P(Y < X)\) in the bivariate exponential model, Communications in Statistics—Theory and Methods, 10, 2515-2524, (1981)
[6] Baklizi, A., Likelihood and Bayesian estimation of \(P r(X < Y)\) using lower record values from the generalized exponential distribution, Computational Statistics and Data Analysis, 52, 3468-3473, (2008) · Zbl 1452.62722
[7] Balakrishnan, N.; Leung, M. Y., Order statistics from the type I generalized logistic distribution, Communications in Statistics—Simulation and Computation, 17, 1, 25-50, (1988) · Zbl 0695.62018
[8] Bamber, D., The area above the ordinal dominance graph and the area below the receiver operating graph, Journal of Mathematical Psychology, 12, 387-415, (1975) · Zbl 0327.92017
[9] Chen, M. H.; Shao, Q. M., Monte Carlo estimation of Bayesian credible and HPD intervals, Journal of Computational and Graphical Statistics, 8, 69-92, (1999)
[10] Congdon, P., Bayesian statistical modeling, (2001), John Wiley New York
[11] Efron, B., (The Jackknife, the Bootstrap and Other Re-Sampling Plans, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 34, (1982), SIAM Philadelphia, PA)
[12] Gradshteyn, I. S.; Ryzhik, I. M., Table of integrals, series, and products, (2007), Academic Press San Diego · Zbl 1208.65001
[13] Gupta, R. D.; Gupta, R. C., Estimation of \(P(a_0 X > b_0 Y)\) in the multivarite normal case, Statistics, 1, 91-97, (1990)
[14] Gupta, R. D.; Kundu, D., Generalized logistic distributions, Journal of Applied Statistical Sciences, 18, 51-66, (2010)
[15] Hall, I. J., Approximate one-sided tolerance limits for the difference or sum of two independent normal variates, Journal of Quality Technology, 16, 15-19, (1984)
[16] Hall, P., Theoretical comparison of bootstrap confidence intervals, Annals of Statistics, 16, 927-953, (1988) · Zbl 0663.62046
[17] Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous univariate distributions, vol. 2, (1995), Wiley and Sons New York · Zbl 0821.62001
[18] Kotz, S.; Lumelskii, Y.; Pensky, M., The stress-strength model and its generalizations, (2003), World Scientific New York · Zbl 1017.62100
[19] Kundu, D.; Gupta, R. D., Estimation of \(P(Y < X)\) for the generalized exponential distribution, Metrika, 61, 291-308, (2005) · Zbl 1079.62032
[20] Kundu, D.; Gupta, R. D., Estimation of \(P(Y < X)\) for Weibull distributions, IEEE Transcations on Reliability, 55, 2, 270-280, (2006)
[21] Lawless, J. F., Statistical models and methods for lifetime data, (2003), John Wiley and Sons New York · Zbl 1015.62093
[22] Mukherjee, S. P.; Maiti, S. S., Stress-strength reliability in the Weibull case, (Frontiers in Reliability, vol. 4, (1998), World Scientific Singapore), 231-248 · Zbl 0926.62098
[23] Nelson, W., Applied life data analysis, (1982), John Wiley and Sons New York · Zbl 0579.62089
[24] Ragab, A., Estimation and predictive density for the generalized logistic distribution, Microelectronics and Reliability, 31, 91-95, (1991)
[25] Raqab, M. Z.; Madi, T.; Kundu, D., Estimation of \(P(Y < X)\) for the three-parameter generalized exponential distribution, Communications in Statistics—Theory and Methods, 37, 2854-2865, (2008) · Zbl 1292.62041
[26] Rezaei, S.; Tahmasbi, R.; Mahmoodi, B. M., Estimation of \(P(Y < X)\) for generalized Pareto distribution, Journal of Statistical Planning and Inference, 140, 480-494, (2010) · Zbl 1177.62024
[27] Saracoglua, B.; Kinacia, I.; Kundu, D., On estimation of \(R = P(Y < X)\) for exponential distribution under progressive type-II censoring, Journal of Statistical Computation and Simulation, 82, 5, 729-744, (2012) · Zbl 1431.62463
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.