Tests of goodness of fit based on the L-2-Wasserstein distance

Ne consider the Wasserstein distance between a sample distribution and the set of normal distributions as a measure of nonnormality. By considering th e standardized version of this distance me obtain a version of Shapiro-Wilk 's test of normality. The asymptotic behavior of the statistic is studied u sing approximations of the quantile process by Brownian bridges. This metho d differs from the "ad hoc" method of de Wet and Venter and permits a simil ar analysis for testing other location scale families.