3 research outputs found
Large-dimensional behavior of regularized Maronna's M-estimators of covariance matrices
Robust estimators of large covariance matrices are considered, comprising
regularized (linear shrinkage) modifications of Maronna's classical
M-estimators. These estimators provide robustness to outliers, while
simultaneously being well-defined when the number of samples does not exceed
the number of variables. By applying tools from random matrix theory, we
characterize the asymptotic performance of such estimators when the numbers of
samples and variables grow large together. In particular, our results show
that, when outliers are absent, many estimators of the regularized-Maronna type
share the same asymptotic performance, and for these estimators we present a
data-driven method for choosing the asymptotically optimal regularization
parameter with respect to a quadratic loss. Robustness in the presence of
outliers is then studied: in the non-regularized case, a large-dimensional
robustness metric is proposed, and explicitly computed for two particular types
of estimators, exhibiting interesting differences depending on the underlying
contamination model. The impact of outliers in regularized estimators is then
studied, with interesting differences with respect to the non-regularized case,
leading to new practical insights on the choice of particular estimators.Comment: 15 pages, 6 figure
Large dimensional analysis of Maronna's M-estimator with outliers
International audienceBuilding on recent results in the random matrix analysis of robust estimators of scatter, we show that a certain class of such estimators obtained from samples containing outliers behaves similar to a well-known random matrix model in the limiting regime where both the population and sample sizes grow to infinity at the same speed. This result allows us to understand the structure of such estimators when a certain fraction of the samples is corrupted by outliers and, in particular, to derive their asymptotic eigenvalue distributions. This analysis is a first step towards an improved usage of robust estimation methods under the presence of outliers when the number of independent observations is not too large compared to the size of the population
Large Dimensional Analysis of Robust M-estimators of Covariance with Outliers
A large dimensional characterization of robust M-estimators of covariance (or
scatter) is provided under the assumption that the dataset comprises
independent (essentially Gaussian) legitimate samples as well as arbitrary
deterministic samples, referred to as outliers. Building upon recent random
matrix advances in the area of robust statistics, we specifically show that the
so-called Maronna M-estimator of scatter asymptotically behaves similar to
well-known random matrices when the population and sample sizes grow together
to infinity. The introduction of outliers leads the robust estimator to behave
asymptotically as the weighted sum of the sample outer products, with a
constant weight for all legitimate samples and different weights for the
outliers. A fine analysis of this structure reveals importantly that the
propensity of the M-estimator to attenuate (or enhance) the impact of outliers
is mostly dictated by the alignment of the outliers with the inverse population
covariance matrix of the legitimate samples. Thus, robust M-estimators can
bring substantial benefits over more simplistic estimators such as the
per-sample normalized version of the sample covariance matrix, which is not
capable of differentiating the outlying samples. The analysis shows that,
within the class of Maronna's estimators of scatter, the Huber estimator is
most favorable for rejecting outliers. On the contrary, estimators more similar
to Tyler's scale invariant estimator (often preferred in the literature) run
the risk of inadvertently enhancing some outliers.Comment: Submitted to IEEE Transactions on Signal Processin