Nonparametric multivariate kurtosis and tailweight measures

For nonparametric exploration or description of a distribution, the treatment of location, spread, symmetry, and skewness is followed by characterization of kurtosis. Classical moment-based kurtosis measures the dispersion of a distribution about its 'shoulders'. Here, we consider quantile-based kurtosis measures. These are robust, defined more widely, and discriminate better among shapes. A univariate quantile-based kurtosis measure of [Groeneveld, R.A. and Meeden, G., 1984, Measuring skewness and kurtosis. The Statistician, 33, 391-399.] is extended to the multivariate case by representing it as a transform of a dispersion functional. A family of such kurtosis measures defined for a given distribution and taken together comprises a real-valued 'kurtosis functional', which has intuitive appeal as a convenient two-dimensional curve for description of the kurtosis of the distribution. Several multivariate distributions in any dimension may thus be compared with respect to their kurtosis in a single two-dimensional plot. Important properties of the new multivariate kurtosis measures are established. For example, for elliptically symmetric distributions, this measure determines the distribution within affine equivalence. Related tailweight measures, influence curves, and asymptotic behavior of sample versions are also discussed.