Estimation of the correlation coefficient based on selected data

In psychometrics, if is often the case that one encounters data that may not be considered random but selected in a systematic way according to some explanatory variable. In this article, maximum likelihood estimation is considered when data are supposed to arise from a bivariate normal distribution that is truncated in an extreme way. Two methods are presented and compared, one of them being purely numerical, while the other is based on an approximation. Both methods are tried on both simulated and on real data. The purely numerical method is shown to be the most reliable over all, but in some cases, the computationally less burdensome approximate method turns out to work almost as well.