Maximum likelihood estimation of the correlation coefficient in a bivariate normal model with missing data

The maximum likelihood estimator (MLE) of the correlation coefficient and its asymptotic properties are well-known for bivariate normal data when no observations are missing. The situation in which one of the two variates is not observed in some of the data is examined herein. The MLE of the correlation coefficient in the missing data case is shown to be asymptotically normal with variance smaller than that when using only the complete bivariate data. The asymptotic results are valid when the number of incomplete data pairs tends to infinity at a faster rate than the number of complete data pairs. The proofs require some mathematical detail since the sample sizes tend to infinity simultaneously rather than individually. A small simulation study corroborates the theoretical results.