Bivariate general exponential models with stress-strength reliability application
Tóm tắt
In this paper, we introduce two families of general bivariate distributions. We refer to these families as general bivariate exponential family and general bivariate inverse exponential family. Many bivariate distributions in the literature are members of the proposed families. Some properties of the proposed families are discussed, as well as a characterization associated with the stress-strength reliability parameter, R, is presented. Concerning R, the maximum likelihood estimators and a simple estimator with an explicit form depending on some marginal distributions are obtained in case of complete sampling. When the stress is censored at the strength, an explicit estimator of R is also obtained. The results obtained can be applied to a variety of bivariate distributions in the literature. A numerical illustration is applied on some well-known distributions. Finally a real data example is presented to fit one of the proposed models.
Tài liệu tham khảo
Mokhlis, N.A., Ibrahim, E.J., Gharieb, M.D.: Stress-strength reliability with general form distributions. Commun. Stat. Theory. Methods. 46(3), 1230–1246 (2017)
Mokhlis, N.A.: Reliability of a stress-strength model with Burr type III distributions. Commun. Stat. Theory. Methods. 34(7), 1643–1657 (2005)
Kundu, D., Gupta, R.D.: Estimation of P(Y<X) for Weibull distribution. IEEE. Trans. Reliability. 55, 270–280 (2006)
Singh, S.K., Singh, U., Singh Yadav, A., Vishwkarma, P.K.: On the estimation of stress-strength reliability parameter of inverted exponential distribution. Int. J. Sci. World. 3, 98–112 (2015)
Kotz, S., Lumelskii, Y., Pensky, M.: The Stress-Strength Model and Its Generalizations: Theory and Applications. World Scientific, Singapore (2003)
Mokhlis, N.M.: Reliability of strength model with a bivariate exponential distribution. J. Egypt. Math. Soc. 14, 69–78 (2006)
Nadarjah, S., Kotz, S.: Reliability for some bivariate exponential distributions. Math. Probl. Eng. 2006, 1–14 (2006)
Nguimkeu, P., Rekkas, M., Wong, A.: Interval estimation for the stress-strength reliability with bivariate normal variables. Open J. Stat. 4, 630–640 (2014)
Pak, A., Khoolenjani, N.B., Jafari, A.A.: Inference on P (Y< X) in bivariate Rayleigh distribution. Commun. Stat. Theory. Methods. 43(22), 4881–4892 (2014)
Abdel-Hamid, A.H.: Stress-strength reliability for general bivariate distributions. J. Egypt. Math. Soc. 5, 617–621 (2016)
Kolesárová, A., Mesiar, R., Saminger-Platz, S.: Generalized Farlie-Gumbel-Morgenstern Copulas, International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, pp. 244–252. Springer, Cham (2018)
Arnold, B.C., Arvanitis, M.A.: On bivariate pseudo-exponential distributions. J. Appl. Stat. 1–13 (2019). https://doi.org/10.1080/02664763.2019.1686132
El-Bassiouny, A.H., Shahen, H.S., Abouhawwash, M.: A new bivariate modified Weibull distribution and its extended distribution. J. Stat. Appl. Probability. 7(2), 217–231 (2018)
Sarhan, A.: The bivariate generalized Rayleigh distribution. J. Math. Sci. Mod. 2(2), 99–111 (2019)
Marshall, A.W., Olkin, I.: A multivariate exponential distribution. J. Am. Stat. Assoc. 62, 30–44 (1967)
Hanagel, D.: Estimation of reliability when stress is censored at strength. Commun. Stat. Theory. Methods. 26(4), 911–919 (1997)
Kotz, S., Balakrishnan, N., Johnson, N.L.: Continuous Multivariate Distributions, vol. 1. Models and Applications, 1, 2nd ed., John Wiley & Sons, New York (2004)
Csörgő, S., Welsh, A.H.: Testing for exponential and Marshall–Olkin distributions. J. Stat. Plann. Inference. 23(3), 287–300 (1989)
Kundu, D., Gupta, R.D.: Modified Sarhan-Balakrishnan singular bivariate distribution. J. Stat. Plann. Inference. 140(2), 526–538 (2010)
Jamalizadeh, A., Kundu, D.: Weighted Marshall–Olkin bivariate exponential distribution. Statistics. 47(5), 917–928 (2013)