Structural properties and convergence results for contours of sample statistical depth functions

Statistical depth functions have become increasingly used in nonparametric inference for multivariate data. Here the contours of such functions are st udied. Structural properties of the regions enclosed by contours, such as a ffine equivariance, nestedness, connectedness and compactness, and almost s ure convergence results for sample depth contours, are established. Also, s pecialized results are established for some popular depth functions, includ ing halfspace depth, and for the case of elliptical distributions. Finally, some needed foundational results on almost sure convergence of sample dept h functions are provided.