Generalized Data Depth with Modern Statistics Application

Tukey's depth offers a powerful tool for nonparametric inference and estimation but also encounters serious computational and methodological difficulties in modern statistical data analysis. This article studies how to generalize Tukey's depth to problems defined in a restricted space that may be curved or have boundaries, and to problems with a non-differentiable objective. First, using a manifold approach, we propose a broad class of Riemannian depth for smooth problems defined on a Riemannian manifold and showcase its applications in spherical data analysis, and principal component analysis, and multivariate orthogonal regression. Moreover, for nonsmooth problems, we introduce additional slack variables and inequality constraints to define a novel slacked data depth, which can perform center-outward rankings of estimators arising from sparse learning and reduced rank regression. Real data examples illustrate the usefulness of some proposed data depths.