On the limiting distribution of and critical values for an origin-invariant bivariate Cramer--von Mises-type statistic

The bivariate Cramer--von Mises statistic has been proposed as a basis for goodness-of-fit testing for continuous bivariate distributions. This statistic can be expressed as the integral, over the unit square I2, of the square of the bivariate uniform empirical process. For some time it apparently went unnoticed that this statistic's value is not 'origin-invariant', i.e., its value depends on which corner of I2 is identified as the origin. Zimmerman (1993) proposed a modification of the bivariate Cramer--von Mises statistic that does not suffer from this defect. In this paper this statistic's limiting null distribution is derived and tabulated using the principal component decomposition method.