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CPS1085 Manoj C.
4. AILs of HPD intervals are smaller and the associated CPs are higher
than that of bootstrap confidence intervals.

3.2 Illustration using real data
We also illustrate an application for the inferential procedures on =
( < ) developed in the previous sections. For that, we consider an
example given in Hoffman and Johnson (2015) on water quality level of
freshwater streams across the commonwealth of Virginia in USA. We have
obtained the MLE and Bayes estimates of P(Y < X) based on first r order
statistics of observation and its concomitants on variates and shows that
the estimated values of ( < ) is more or less 0.9 for all the cases.
Therefore we can claim that the concentration level of Zinc in freshwater across
Virginia is relatively lower than the worldwide quality standard of zinc in water.

4. Conclusion
In this work, we have considered the problem of estimation of = ( <
) for bivariate normal distribution using type II censored order statistics and
its concomitants. The maximum likelihood and Bayes estimators have been
obtained for . For obtaining the Bayes estimates, importance sampling
method has been applied. Based on the simulation study we have the
following conclusions. The bias and MSE of all estimators decrease when the
number of uncensored observations r increases. Bayes estimators perform
better than that of MLEs in terms of bias and MSE. Among the Bayes
estimators, estimator under SEL function perform better when ≠ and
2
1
estimator under EL function perform better when = . AILs of HPD
2
1
intervals are smaller and the associated CPs are higher than that of bootstrap
confidence intervals.

References
1. Abu-Salih, M. S., Shamseldin, A. A.(1988) Bayesian estimation of P(X < Y )
for bivariate exponential distribution, Arab Gulf J. Sci. Res. A. Math. Phys.
Sci., 6(1), 17-26.
2. Awad, A., Azzam, M., Hamdan, M. ( 1981 ) Some inference results on P(Y
< X) in the bivariate exponential model, Communications in Statistics
Theory and Methods, 10, 2515-2525.
3. Chen M.H., Shao Q.M. (1999) Monte Carlo estimation of Bayesian credible
and HPD intervals, Journal of Computational and Graphical Statistics, 8,
69-92.
4. Cramer, E. (2001) Inference for stress-strength models based on wienman
multivariate exponential samples, Communications in Statistics - Theory
and Methods, 30, 331-346

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