A Depth Function and a Scale Curve Based on Spatial Quantiles

Spatial quantiles, based on the L1 norm in a certain sense, provide an appealing vector extension of univariate quantiles and generate a useful “volume” functional based on spatial “central regions” of increasing size. A plot of this functional as a “spatial scale curve” provides a convenient twodimensional characterization of the spread of a multivariate distribution of any dimension. We discuss this curve and establish weak convergence of the empirical version. As a tool, we introduce and study a new statistical depth function which is naturally associated with spatial quantiles. Other depth functions that generate L1-based multivariate quantiles are also noted.