Abstract
In this paper, we consider the problem of estimation of \(R=P(X<Y)\), when X and Y are dependent, using bivariate ranked set sampling. The maximum likelihood estimates (MLEs) and Bayes estimates (BEs) of R are obtained based on ranked set sample when (X,Y) follows bivariate normal distribution. BEs are obtained based on both symmetric and asymmetric loss functions. The percentile bootstrap and HPD confidence intervals for R are also obtained. Simulation studies are carried out to find the accuracy of the proposed estimators. A real data is also used to illustrate the inferential procedures developed in this paper.
Similar content being viewed by others
References
Abu-Salih, M.S., Shamseldin, A.A.: Bayesian estimation of \(P(X < Y)\) for bivariate exponential distribution. Arab Gulf J. Sci. Res. A. Math. Phys. Sci. 6(1), 17–26 (1988)
Akgül, F.G., Şenoğlu, B.: Estimation of \(\text{ P }(\text{ X } < \text{ Y })\) using ranked set sampling for the Weibull distribution. Qual. Technol. Quant. Manag. 14(3), 296–309 (2017)
Akgül, F.G., AcıtaŞ, S., Şenoğlu, B.: Inferences on stress-strength reliability based on ranked set sampling data in case of Lindley distribution. J. Stat. Comput. Simul. 88(15), 3018–3032 (2018)
Awad, A., Azzam, M., Hamdan, M.: Some inference results on \(P(Y < X)\) in the bivariate exponential model. Commun. Stat. Theory Methods 10, 2515–2525 (1981)
Balakrishnan, N., Kim, J.A.: Point and interval estimation for bivariate normal distribution based on progressively type-II censored data. Commun. Stat. Theory Methods 34, 1297–1347 (2005)
Chen, M.H., Shao, Q.M.: Monte Carlo estimation of Bayesian credible and HPD intervals. J. Comput. Graph. Stat. 8, 69–92 (1999)
Chen, Z., Bai, Z., Sinha, B.K.: Ranked Set Sampling, Theory and Applications. Springer, New York (2004)
Cramer, E.: Inference for stress-strength models based on wienman multivariate exponential samples. Commun. Stat. Theory Methods 30, 331–346 (2001)
Dong, X., Zhang, L., Li, F.: Estimation of reliability for exponential distributions using ranked set sampling with unequal samples. Qual. Technol. Quant. Manag. 10, 319–328 (2013)
Enis, P., Geisser, S.: Estimation of the probability that \(Y<X\). J. Am. Stat. Assoc. 66, 162–186 (1971)
Gill, P.S., Tiku, M.L., Vaughan, D.C.: Inference problems in life testing under multivariate normality. J. Appl. Stat. 17, 133–147 (1990)
Hanagal, D.D.: Testing reliability in a bivariate exponential stress-strength model. J. Indian Stat. Assoc. 33, 41–45 (1995)
Hanagal, D.D.: Estimation of reliability when stress is censored at strength. Commun. Stat. Theory Methods 26(4), 911–919 (1997)
He, Q., Nagaraja, H.N.: Correlation estimation in Downton’s bivariate exponential distribution using incomplete samples. J. Stat. Comput. Simul. 81, 531–546 (2011)
Hoffman, H.J., Johnson, R.E.: Pseudo-likelihood estimation of multivariate normal parameters in the presence of left-censored data. J. Agric. Biol. Environ. Stat. 20, 156–171 (2015)
Hussian, M.A.: Estimation of stress-strength model for generalized inverted exponential distribution using ranked set sampling. Int. J. Adv. Eng. Technol. 6, 2354–2362 (2014)
Jana, P.K.: Estimation of \(P(Y<X)\) in the bivariate exponential case due to Marshall–Olkin. J. Indian Stat. Assoc. 32, 35–37 (1994)
Jana, P.K., Roy, D.: Estimation of reliability under stress-strength model in a bivariate exponential set-up. Calcutta Stat. Assoc. Bull. 44, 175–181 (1994)
McIntyre, G.A.: A method for unbiased selective sampling using ranked sets. Aust. J. Agric. Res. 3, 385–390 (1952)
Mukherjee, S.P., Saran, L.K.: Estimation of failure probability from a bivariate normal stress-strength distribution. Microelectron. Reliab. 25, 692–702 (1985)
Muttlak, H.A., Abu-Dayyah, W.A., Saleh, M.F., Al-Sawi, E.: Estimating \(P(Y < X)\) using ranked set sampling in case of the exponential distribution. Commun. Stat. Theory Methods 39, 1855–1868 (2010)
Nadarajah, S., Kotz, S.: Reliability for some bivariate exponential distributions. Math. Probl. Eng. 2006, 1–14 (2006). (Article ID 41652)
Sengupta, S., Mukhuti, S.: Unbiased estimation of \(P(X < Y)\) using ranked set sample data. Statistics 42, 223–230 (2008)
Tokdar, S.T., Tass, R.E.: Importance sampling: a review. Wiley Interdiscip. Rev. Comput. Stat. 2, 54–60 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chacko, M., Mathew, S. Inference on \(P(X<Y)\) for bivariate normal distribution based on ranked set sample. METRON 77, 239–252 (2019). https://doi.org/10.1007/s40300-019-00154-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40300-019-00154-5