Uniform inference and differentiation under perturbations of the testing body

Develop uniform inference for entire depth contours and derive differentiability results for depth regions under perturbations of the testing body, under suitable additional nondegeneracy assumptions.

Background

The paper establishes uniform consistency for empirical $K$-depth functions and Hausdorff consistency for individual depth regions at regular levels, assuming a fixed testing body. It also proves deterministic stability of depth functions and regular depth regions when the testing body is perturbed in the Hausdorff metric.

The authors identify broader unresolved tasks: extending inference from pointwise or fixed-level results to entire depth contours, and establishing differentiability of depth regions as the testing body varies. They note that such developments would require additional nondegeneracy assumptions and are not addressed in the paper.