Generalized resilience and robust statistics

Robust statistics traditionally focuses on outliers, or perturbations in total variation distance. However, a dataset could be maliciously corrupted many other ways, such as systematic measurement errors and missing covariates. We consider corruption either TV Wasserstein distance, show that robust estimation is possible whenever the true population distribution satisfies property called generalized resilience, which holds under moment hypercontractive conditions. For model, our finite-sample analysis improves over previous results for mean with bounded kth moment, linear regression, joint covariance estimation. W1 corruption, we provide first guarantees second regression. Technically, estimators are generalization of minimum distance (MD) functionals, project onto given set well-behaved distributions. The error these MD functionals by certain modulus continuity, method upper bounding this class resilient distributions, usually gives sharp population-level good guarantees.