Estimation of P(X < Y ) for Morgenstern Type Bivariate Exponential Distribution Based on Ranked Set Sample

Shiny  Mathew; Manoj  Chacko

Authors

Shiny Mathew Department of Statistics, University of Kerala, Trivandrum, Kerala, India
Manoj Chacko Department of Statistics, University of Kerala, Trivandrum, Kerala, India

Keywords:

Ranked set sampling, Morgenstern type bivariate exponential distribution, maximum likelihood estimator, Bayesian estimation

Abstract

Estimation of R = P(X < Y ) has been intensively investigated in the literature using parametric and nonparametric approaches under different sampling schemes when X and Y are independent random variables. In this paper, we consider the problem of estimation of R when X and Y are dependent random variables based on ranked set sample (RSS). The maximum likelihood estimates (MLEs) and Bayes estimates (BEs) of R are obtained based on RSS when (X, Y ) follows Morgenstern type bivariate exponential 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.

References

Akgül, F. G., & Şenoğlu, B. (2017). Estimation of P(X

Chacko, M., & Thomas, P. Y. (2008). Estimation of a parameter of Morgenstern type bivariate exponential distribution by ranked set sampling. Annals of the Institute of Statistical Mathematics, 60(2), 301–318.

Chacko, M. (2017). Bayesian estimation based on ranked set sample from Morgenstern type bivariate exponential distribution when ranking is imperfect. Metrika, 80(3), 333–349.

Chacko, M., & Mathew, S. (2019). Inference on P(X

Chacko, M., & Mathew, S. (2022). Inference on P(X

Chen, Z., Bai, Z., & Sinha, B. K. (2004). Ranked set sampling: Theory and applications. New York: Springer.

Chen, M.-H., & Shao, Q.-M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69–92.

Dong, X., Zhang, L., & Li, F. (2013). Estimation of reliability for exponential distributions using ranked set sampling with unequal samples. Quality and Technology & Quantitative Management, 10(3), 319–328.

Hanagal, D. D. (2011). Modelling survival data using frailty models. New York: CRC Press.

Hanagal, D. D., & Sharma, R. (2015). Bayesian inference in Marshall–Olkin bivariate exponential shared gamma frailty regression model under random censoring. Communications in Statistics—Theory and Methods, 44(1), 24–47.

Huster, W. J., Brookmeyer, R., & Self, S. G. (1989). Modelling paired survival data with covariates. Biometrics, 45(1), 145–156.

Hussian, M. A. (2014). Estimation of stress–strength model for generalized inverted exponential distribution using ranked set sampling. International Journal of Advanced Engineering Technology, 6(6), 2354–2362.

Kotz, S., Balakrishnan, N., & Johnson, N. L. (2000). Distributions in statistics: Continuous multivariate distributions. New York: John Wiley & Sons.

Kotz, S., Lumelskii, Y., & Pensky, M. (2003). The stress–strength model and its generalizations: Theory and applications. Singapore: World Scientific.

Lai, C. D., & Balakrishnan, N. (2009). Continuous bivariate distributions. New York: Springer.

McIntyre, G. A. (1952). A method for unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research, 3(4), 385–390.

Muttlak, H. A., Abu-Dayyah, W. A., Saleh, M., & Al-Sawi, E. (2010). Estimating

P(Y

Nadarajah, S., & Kotz, S. (2006). Reliability for some bivariate exponential distributions. Mathematical Problems in Engineering, Article ID 41652, 1–14.

Sengupta, S., & Mukhuti, S. (2008). Unbiased estimation of P(X

Tokdar, S. T., & Kass, R. E. (2010). Importance sampling: A review. WIREs Computational Statistics, 2(1), 54–60.