It seems fair that the usual sample correlation coefficient should allow improvement when additional samples from the marginals are available. However, some intuitive attempts fail badly. Using control variates, a simple method is presented which asymptotically achieves the optimal improvement.

The bivariate normal distribution function is approximated with emphasis on situations where the correlation coefficient is large. The high accuracy of the approximation is illustrated by numerical examples. Moreover, exact upper and lower bounds are presented as well as asymptotic results on...