Pseudo maximum likelihood estimates using ranked set sampling with applications to estimating correlation

Suppose θ is the parameter of interest and λ is the nuisance parameter. When obtaining the maximum likelihood estimator (MLE) of θ in the presence of λ requires intensive computation, the pseudo MLE of θ, based on a pseudo likelihood function, can be used. Gong and Samaniego (Ann. Stat. 9:861–869, 1981) proposed a pseudo MLE (PMLE) based on simple random samples. Ranked set sampling has been applied to the bivariate variables (X,Y) where measuring one of the variables is difficult or costly. In this paper, we obtain the pseudo MLE of the correlation coefficient from a bivariate normal distribution (X,Y) based on ranked set samples, assuming that Y is difficult or more expensive to measure and that the mean and variance of Y are the nuisance parameters. The PMLE is compared with three other estimators of the correlation coefficient. Simulations show that the PMLE is more (less) efficient than other estimators, depending on value of ρ. Testing of soil contamination provides an example of the use of the methods.