Extreme smoothing and testing for multivariate normality

Recently, Bowman and Foster (1993) proposed to base a test for multivariate normality on a L$^2$ distance between a nonparametric kernel density estimator and the parametric density estimator under normality, applied to the empirically standardized data. We show that, for a fixed bandwidth (not depending on the sample size), the test of Bowman and Foster is a member of the class of invariant and universally consistent procedures suggested by Henze and Zirkler (1990). Moreover, we identify and study the tests for multivariate normality obtained by letting the bandwidth tend to zero and to infinity. While the former test statistic is based solely on the Euclidean norm of the standardized data, letting the bandwidth tend to infinity yields a weighted sum of Mardia's time-honoured measure of multivariate skewness and a sample version of a recently introduced skewness measure of Móri, Rohatgi and Székely (1993).