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Stress-strength based on \(m\)-generalized order statistics and concomitant for dependent families. (English) Zbl 07088750
Summary: The stress-strength model is proposed based on the \(m\)-generalized order statistics and the corresponding concomitant. For the dependency between \(m\)-generalized order statistics and its concomitant, a bivariate copula expansion is considered and the stress-strength model is obtained for two special cases of order statistics and upper record values. In the particular case of copula function, the generalized Farlie-Gumbel-Morgenstern bivariate distribution function is considered with proportional reversed hazard functions as marginal functions. Based on the order statistics and record values, two estimators of stress-strength are presented using a procedure similar to the inference functions for margins. Finally, a simulation study is carried out which shows the good performance of the proposed estimators for a finite sample.

MSC:
62G30 Order statistics; empirical distribution functions
62N05 Reliability and life testing
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[1] Adimari, G.; Chiogna, M., Partially parametric interval estimation of Pr\(\{Y>X\}\), Comput. Stat. Data Anal. 51 (2006), 1875-1891
[2] Al-Mutairi, D. K.; Ghitany, M. E.; Kundu, D., Inferences on stress-strength reliability from weighted Lindley distributions, Commun. Stat., Theory Methods 44 (2015), 4096-4113
[3] Asgharzadeh, A.; Valiollahi, R.; Raqab, M. Z., Stress-strength reliability of Weibull distribution based on progressively censored samples, SORT 35 (2011), 103-124
[4] Bairamov, I.; Kotz, S.; Bekçi, M., New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics, J. Appl. Stat. 28 (2001), 521-536
[5] Baklizi, A., Estimation of Pr\((X < Y)\) using record values in the one and two parameter exponential distributions, Commun. Stat., Theory Methods 37 (2008), 692-698 corrigendum ibid. 40 2011 4322-4323
[6] Baklizi, A., Inference on Pr\((X < Y)\) in the two-parameter Weibull model based on records, ISRN Probab. Stat. 2012 (2012), Article ID 263612, 11 pages
[7] Basirat, M.; Baratpour, S.; Ahmadi, J., Statistical inferences for stress-strength in the proportional hazard models based on progressive Type-II censored samples, J. Stat. Comput. Simulation 85 (2015), 431-449
[8] Beg, M. I.; Ahsanullah, M., Concomitants of generalized order statistics from Farlie-Gumbel-Morgenstern distributions, Stat. Methodol. 5 (2008), 1-20
[9] Bose, A.; Gangopadhyay, S., A note on concomitants of records, Stat. Methodol. 10 (2013), 103-112
[10] Chacko, M.; Thomas, P. Y., Estimation of parameters of bivariate normal distribution using concomitants of record values, Stat. Pap. 49 263-275 (2008)
[11] Condino, F.; Domma, F.; Latorre, G., Likelihood and Bayesian estimation of \(P(Y>X)\) using lower record values from a proportional reversed hazard family, Stat. Pap. 59 (2018), 467-485
[12] Dagum, C., The generation and distribution of income. The Lorenz curve and the Gini ratio, Economie Appliqu{é}e 33 (1980), 327-367
[13] David, H. A.; Nagaraja, H. N., Order Statistics, Wiley Series in Probability and Statistics, John Wiley & Sons, Chichester (2003)
[14] Dengler, B., On the Asymptotic Behaviour of the Estimator of Kendall’s Tau, Ph.D. Thesis, TU Vienna (2010)
[15] Domma, F.; Giordano, S., A stress-strength model with dependent variables to measure household financial fragility, Stat. Methods Appl. 21 (2012), 375-389
[16] Domma, F.; Giordano, S., A copula-based approach to account for dependence in stress-strength models, Stat. Pap. 54 (2013), 807-826
[17] Domma, F.; Giordano, S., Concomitants of \(m\)-generalized order statistics from generalized Farlie-Gumbel-Morgenstern distribution family, J. Comput. Appl. Math. 294 (2016), 413-435
[18] Farlie, D. J. G., The performance of some correlation coefficients for a general bivariate distribution, Biometrika 47 (1960), 307-323
[19] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products, Elsevier/ Press, Amsterdam (2007)
[20] Gupta, R. C.; Subramanian, S., Estimation of reliability in a bivariate normal distribution with equal coefficients of variation, Commun. Stat., Simulation Comput. 27 (1998), 675-698
[21] Hanagal, D. D., Estimation of reliability when stress is censored at strength, Commun. Stat., Theory Methods 26 (1997), 911-919
[22] Jaheen, Z. F., Empirical Bayes inference for generalized exponential distribution based on records, Commun. Stat., Theory Methods 33 (2004), 1851-1861
[23] Joe, H., Multivariate Models and Multivariate Dependence Concepts, Monographs on Statistics and Applied Probability 73, Chapman & Hall, London (1997)
[24] Joe, H.; Xu, J. J., The estimation method of inference functions for margins for multivariate models, Technical Report #166, University of British Columbia, Vancouver (1996)
[25] Kamps, U., A concept of generalized order statistics, J. Stat. Plann. Inference 48 (1995), 1-23
[26] Kotz, S.; Lumelskii, Y.; Pensky, M., The Stress-Strength Model and Its Generalizations. Theory and Applications, World Scientific, River Edge (2003)
[27] Marshall, A. W.; Olkin, I., Life Distributions. Structure of Nonparametric, Semiparametric, and Parametric Families, Springer Series in Statistics, Springer, New York (2007),\99999DOI99999 10.1007/978-0-387-68477-2 \hyphenation{McGraw}
[28] Mood, A. M.; Graybill, F. A.; Boes, D. C., Introduction to the Theory of Statistics, McGraw-Hill Series in Probability and Statistics, McGraw-Hill Book Company, New York (1974)
[29] Nadar, M.; Kızılaslan, F., Classical and Bayesian estimation of \(P(X < Y)\) using upper record values from Kumaraswamy’s distribution, Stat. Pap. 55 (2014), 751-783
[30] Nadarajah, S., Reliability for some bivariate beta distributions, Math. Probl. Eng. 2005 (2005), 101-111
[31] Nadarajah, S., Expansions for bivariate copulas, Stat. Probab. Lett. 100 (2015), 77-84
[32] Nelsen, R. B., An Introduction to Copulas, Springer Series in Statistics, Springer, New York (2006)
[33] Nevzorov, V. B.; Ahsanullah, M., Some distributions of induced records, Biom. J. 42 (2000), 1069-1081
[34] Pakdaman, Z.; Ahmadi, J., Stress-strength reliability for \(P(X_{r:n_1} < Y_{k:n_2})\) in the exponential case, İstatistik 6 (2013), 92-102
[35] Raqab, M. Z.; Madi, M. T., Inference for the generalized Rayleigh distribution based on progressively censored data, J. Stat. Plann. Inference 141 (2011), 3313-3322
[36] Rezaei, S.; Tahmasbi, R.; Mahmoodi, M., Estimation of \(P[Y>X]\) for generalized Pareto distribution, J. Stat. Plann. Inference 140 (2010), 480-494
[37] Saraçoğlu, B.; Kinaci, I.; Kundu, D., On estimation of \(R=P(Y>X)\) for exponential distribution under progressive type-II censoring, J. Stat. Comput. Simulation 82 (2012), 729-744
[38] Sengupta, S., Unbiased estimation of \(P(Y>X)\) for two-parameter exponential populations using order statistics, Statistics 45 (2011), 179-188
[39] Tahmasebi, S.; Jafari, A. A.; Afshari, M., Concomitants of dual generalized order statistics from Morgenstern type bivariate generalized exponential distribution, J. Stat. Theory Appl. 14 (2015), 1-12
[40] Tarvirdizade, B.; Ahmadpour, M., Estimation of the stress-strength reliability for the two-parameter bathtub-shaped lifetime distribution based on upper record values, Stat. Methodol. 31 (2016), 58-72
[41] Valiollahi, R.; Asgharzadeh, A.; Raqab, M. Z., Estimation of \(P(Y < X)\) for Weibull distribution under progressive Type-II censoring, Commun. Stat., Theory Methods 42 (2013), 4476-4498
[42] Wong, A., Interval estimation of \(P(Y>X)\) for generalized Pareto distribution, J. Stat. Plann. Inference 142 (2012), 601-607
[43] Yang, S. S., General distribution theory of the concomitants of order statistics, Ann. Stat. (1977), 996-1002
[44] Yörübulut, S.; Gebizlioglu, O. L., Bivariate pseudo-Gompertz distribution and concomitants of its order statistics, J. Comput. Appl. Math. 247 (2013), 68-83
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