, Volume 77, Issue 3, pp 239–252 | Cite as

Inference on \(P(X<Y)\) for bivariate normal distribution based on ranked set sample

  • Manoj ChackoEmail author
  • Shiny Mathew


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.


Ranked set sampling Concomitants of order statistics Bayesian estimation Importance sampling 



  1. 1.
    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)MathSciNetzbMATHGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Chen, M.H., Shao, Q.M.: Monte Carlo estimation of Bayesian credible and HPD intervals. J. Comput. Graph. Stat. 8, 69–92 (1999)MathSciNetGoogle Scholar
  7. 7.
    Chen, Z., Bai, Z., Sinha, B.K.: Ranked Set Sampling, Theory and Applications. Springer, New York (2004)CrossRefGoogle Scholar
  8. 8.
    Cramer, E.: Inference for stress-strength models based on wienman multivariate exponential samples. Commun. Stat. Theory Methods 30, 331–346 (2001)CrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Enis, P., Geisser, S.: Estimation of the probability that \(Y<X\). J. Am. Stat. Assoc. 66, 162–186 (1971)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gill, P.S., Tiku, M.L., Vaughan, D.C.: Inference problems in life testing under multivariate normality. J. Appl. Stat. 17, 133–147 (1990)CrossRefGoogle Scholar
  12. 12.
    Hanagal, D.D.: Testing reliability in a bivariate exponential stress-strength model. J. Indian Stat. Assoc. 33, 41–45 (1995)Google Scholar
  13. 13.
    Hanagal, D.D.: Estimation of reliability when stress is censored at strength. Commun. Stat. Theory Methods 26(4), 911–919 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    He, Q., Nagaraja, H.N.: Correlation estimation in Downton’s bivariate exponential distribution using incomplete samples. J. Stat. Comput. Simul. 81, 531–546 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    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)MathSciNetGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    McIntyre, G.A.: A method for unbiased selective sampling using ranked sets. Aust. J. Agric. Res. 3, 385–390 (1952)CrossRefGoogle Scholar
  20. 20.
    Mukherjee, S.P., Saran, L.K.: Estimation of failure probability from a bivariate normal stress-strength distribution. Microelectron. Reliab. 25, 692–702 (1985)CrossRefGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nadarajah, S., Kotz, S.: Reliability for some bivariate exponential distributions. Math. Probl. Eng. 2006, 1–14 (2006). (Article ID 41652) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sengupta, S., Mukhuti, S.: Unbiased estimation of \(P(X < Y)\) using ranked set sample data. Statistics 42, 223–230 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tokdar, S.T., Tass, R.E.: Importance sampling: a review. Wiley Interdiscip. Rev. Comput. Stat. 2, 54–60 (2010)CrossRefGoogle Scholar

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© Sapienza Università di Roma 2019

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of KeralaTrivandrumIndia

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