The concept of statistical depth extends the notions of the median and quantiles to other statistical models. These procedures aim to formalize the idea of identifying deeply embedded fits to a model that are less influenced by contamination. In the multivariate case, Tukey's median was a groundbreaking concept for multivariate location estimation, and its counterpart for scatter matrices has recently attracted considerable interest. The breakdown point and the maximum asymptotic bias are key concepts used to summarize an estimator's behavior under contamination. In the multivariate and regression setting we analyze recently introduced concentration inequalities that provide a unified framework for studying both the statistical convergence rate and robustness of Tukey's median, depth-based scatter matrices and depth-based multivariate regression estimators. We observe that slight variations in these inequalities allow us to visualize the maximum bias behavior of the deepest estimators. We explicitly obtain the maximum bias curve and breakdown point of the deepest scatter matrices. For the location and scale model, we consider two closely related depth formulations, whose deepest estimators display significantly different behavior in terms of breakdown point. A numerical study is performed to compare the finite sample bias performance of several robust estimators in the multivariate setting.
Categories: math.ST stat.TH