Bias robustness of depth estimators in multivariate settings

TL;DR

This paper analyzes the robustness of depth-based multivariate estimators, deriving explicit bias and breakdown points, and compares their finite sample bias performance through numerical studies.

Contribution

It provides explicit formulas for maximum bias, breakdown point, and contamination sensitivity of depth-based estimators, unifies various depth concepts, and evaluates their finite sample robustness.

Findings

Derived maximum bias curves for deepest scatter matrices.

Unified the concept of halfspace depths via residual smallness depth.

Numerical comparison shows differences in finite sample bias among estimators.

Abstract

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. We explicitly obtain the maximum bias curve, contamination sensitivity and breakdown point of the deepest scatter matrices. In the multivariate and regression setting we analyse recently introduced error bounds that provide a unified framework for studying both the statistical convergence rate and robustness of Tukey's median, depth-based scatter matrices and multivariate regression estimators. We observe that slight variations in these inequalities allow us to visualize the maximum bias behavior of the deepest estimators. We also point out that all the halfspace depths under consideration can be obtained from a unifying concept called residual smallness depth. A numerical study is performed to compare the finite sample bias performance of several robust estimators in the multivariate setting.