Nonparametric Multivariate Descriptive Measures Based on Spatial Quantiles

Abstract An appealing way of working with probability distributions, especially in nonparametric inference, is through “descriptive measures” that characterize features of particular interest. One attractive approach is to base the measures on quantiles. Here we consider the multivariate context and utilize the “spatial quantiles”, a recent vector extension of univariate quantiles that is becoming increasingly popular. In terms of these quantiles, we introduce and study nonparametric measures of multivariate location, spread, skewness and kurtosis. In particular, we define a useful “location” functional which augments the well-known “spatial” median and a “volume” functional which plotted as a “spatial scale curve” yields a convenient two-dimensional characterization of the spread of a multivariate distribution of any dimension. These spatial location and volume functionals also play roles in the formulation of “spatial” skewness and kurtosis functionals which reduce to known versions in the univariate case. We also define corresponding spatial “asymmetry” and “kurtosis” curves which are new devices even in the univariate case. Tailweight and peakedness measures, as distinct from kurtosis, are also discussed. To aid better understanding of the spatial quantiles as a foundation for nonparametric multivariate inference and analysis, we also provide some basic perspective on them: their interpretations, properties, strengths and weaknesses.