Optimal allocation for estimating the correlation coefficient of Morgenstern type bivariate exponential distribution by ranked set sampling with concomitant variable

Ranked set sample is applicable whenever ranking of a set of sample units can be done easily by a judgement method of the study variable or of the auxiliary variable. This paper considers ranked set sample based on the auxiliary variable X which is correlated with the study variable Y, where (X, Y) follows Morgenstern type bivariate exponential distribution. The authors discuss the optional allocation for unbiased estimators of the correlation coefficient ρ of the random variables X and Y when the auxiliary variable X is used for ranking the sample units and the study variable Y is measured for estimating the correlation coefficient. This paper first gives a class of unbiased estimators of ρ when the mean θ of the study variable Y is known and obtains an essentially complete subclass of this class. Further, the optimal allocation of the unbiased estimators is found in this subclass and is proved to be Bayes, admissible, and minimax. Finally, the unbiased estimator of ρ under the optimal allocation in the case of known θ is reformed for estimating ρ in the case of unknown θ, and the reformed estimator is shown to be strongly consistent.