Testing for bivariate normality using the empirical distribution function

The problem of testing for bivariate normality using the empirical distribution function is considered. A Cramér-von Mises type statistic is defined and asymptotic percentage points for this statistic given. This involves solving a two-dimensional homogeneous integral equation. Unfortunately the Cramér-von Mises statistic is not invariant under orthogonal transformations of the data so that an invariant statistic is developed. Approximations for the distribution of this statistic are found by Monte Carlo. Applications of the statistics are given. It is shown that the statistics are particularly sensitive to certain kinds of pattern in the data and they could be useful in data analysis apart from providing a formal test of bivariate normality