Robust distances and multivariate outlier detection under heavy tails

Heavy-tailed distributions, such as the Student-t distribution, have long been advocated as “robust” models for multivariate data in many fields. The underlying rationale is that robustness should be achieved by letting the classical maximum-likelihood estimators accommodate extreme observations naturally arising from the process under investigation. However, there is growing recognition that contamination might also occur under non-Gaussian scenarios. In this work we develop a unified approach which exploits high-breakdown estimation to ensure robust estimation of location, scatter and tail parameters under contamination of a multivariate Student-t distribution, when the latter is assumed to rule the data generating process of the uncontaminated part of the data. The framework that allows us to achieve our unified approach is the theory of generalized radius processes. In this framework, we first obtain the influence function of the main statistical functionals associated to robust Mahalanobis distances. We then suggest new statistics for measuring conformance to the multivariate Student-t distribution and an automatic procedure to infer the true values of the degrees of freedom and of the contamination rate. Along our path, we tackle several computational challenges associated to Monte Carlo estimation of the required radius-process quantiles and we provide extensive simulation evidence of the accuracy of our method. We guarantee the replicability of our results and we provide the implementation of the suggested algorithms.