A Vectorial Notion of Skewness and Its Use in Testing for Multivariate Symmetry

By modifying the statistic of Malkovich and Afifi (1973), we introduce and study the properties of a notion of multivariate skewness that provides both a magnitude and an overall direction for the skewness present in multivariate data. This notion leads to a test statistic for the nonparametric null hypothesis of multivariate symmetry. Under mild assumptions, we find the asymptotic distribution of the test statistic and evaluate, by simulation, the convergence of the finite sample size percentiles to their limits. We also present an associated test statistic for multivariate normality.