Calibration of robust covariance estimation under heavy tails
Abstract
The Minimum Covariance Determinant (MCD) estimator is one of the most widely used high-breakdown estimators of multivariate location and scatter. Its practical success relies on suitable corrections, which remove the asymptotic and small-sample biases introduced by trimming. While explicit consistency factors are available under the multivariate normal distribution and, more recently, under multivariate Student-t models, these corrections only address the asymptotic behavior of the estimator. In the Gaussian case, it is known showed that the MCD also exhibits a non-negligible finite-sample bias, particularly for small sample sizes and relatively large dimensions. Such bias propagates to robust Mahalanobis distances and may affect subsequent procedures, including multivariate outlier detection and robust inference. This paper investigates the finite-sample behavior of the MCD estimator and of the related robust distances under multivariate Student-t distributions, a potentially relevant scenario in many application fields. Extensive Monte Carlo experiments are carried out over a broad range of sample sizes, dimensions, trimming levels and degrees of freedom. Based on these experiments, empirical correction factors are obtained and approximated by smooth functions, providing a computationally efficient calibration suitable for practical implementations to heavy-tailed data. The proposed corrections are developed both for the scatter bias and the robust distance quantiles, thus leading to improved approximations for robust inference.
Related
- Robust dimension-free estimation of simple random tensors: optimal guarantees under heavy tails and adversarial contamination
- Residual-based extreme value estimation for heavy tails
- Robust Covariance Estimation for Entry-wise Contaminated and Partially Observed Data via Expectation Maximization Algorithm
- High-Dimensional Regularization of the Spatial Sign Covariance Matrix for Robust Shape Estimation
- Stationary activated random walk: regularity, heavy tails, and remixing
BibTeX
@misc{cerioli2026calibration,
title = {{Calibration of robust covariance estimation under heavy tails}},
author = {Andrea Cerioli and Domenico Perrotta and Francesca Torti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34622},
eprint = {2610.34622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34622},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Calibration of robust covariance estimation under heavy tails
AU - Andrea Cerioli
AU - Domenico Perrotta
AU - Francesca Torti
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34622
UR - https://arxiv.org/abs/2610.34622
AB - The Minimum Covariance Determinant (MCD) estimator is one of the most widely used high-breakdown estimators of multivariate location and scatter. Its practical success relies on suitable corrections, which remove the asymptotic and small-sample biases introduced by trimming. While explicit consistency factors are available under the multivariate normal distribution and, more recently, under multivariate Student-t models, these corrections only address the asymptotic behavior of the estimator. In the Gaussian case, it is known showed that the MCD also exhibits a non-negligible finite-sample bias, particularly for small sample sizes and relatively large dimensions. Such bias propagates to robust Mahalanobis distances and may affect subsequent procedures, including multivariate outlier detection and robust inference. This paper investigates the finite-sample behavior of the MCD estimator and of the related robust distances under multivariate Student-t distributions, a potentially relevant scenario in many application fields. Extensive Monte Carlo experiments are carried out over a broad range of sample sizes, dimensions, trimming levels and degrees of freedom. Based on these experiments, empirical correction factors are obtained and approximated by smooth functions, providing a computationally efficient calibration suitable for practical implementations to heavy-tailed data. The proposed corrections are developed both for the scatter bias and the robust distance quantiles, thus leading to improved approximations for robust inference.
ER -