Inference on correlation from incomplete bivariate samples

For a random sample (X_i,Y_i), 1 ≤ i ≤ n, from an absolutely continuous bivariate population (X,Y), let X_i:n denote the ith X-order statistic and Y_[i:n] be its concomitant. We develop the joint distribution of a Y-order statistic and the Y-concomitant of an X-order statistic, namely that of Y_i:n and Y_[j:n]. The joint distribution can be useful in pursuing the properties of estimators of population correlation coefficient for incomplete bivariate samples.

We obtain expressions for the elements of the limiting Fisher information (FI) matrix for a bivariate Type II right censored sample (X_i:n,Y_[i:n]), 1≤ i≤ r<n. We evaluate these elements for bivariate normal (BVN), Downton's bivariate exponential (DBVE) and Gumbel Type II bivariate exponential (G₂BVE) distributions. We use the expressions for the elements of the FI matrix to evaluate the various correlation estimates from incomplete bivariate samples.

Suppose our incomplete data sets consist of either only the Y values and the ranks of the associated X values or a bivariate Type II right censored sample from (X,Y). We assume (X,Y) has either a BVN or a DBVE distribution with unknown correlation coefficient ρ. We investigate the estimators of ρ based on these two types of incomplete data. For both distributions we use simulation to examine several estimators and obtain their estimated relative efficiencies. For the BVN case, the estimators based only on the concomitants can be highly efficient.

We investigate the estimation of the dependence parameter using a complete bivariate sample and the above two types of incomplete data from the G₂BVE distribution. We also discuss several methods for estimating the correlation coefficient ρ between X and Y under the BVN assumption when the X values below a certain detection limit cannot be observed.