  This article studies two regularized robust estimators of scatter matrices
proposed (and proved to be well defined) in parallel in (Chen et al., 2011) and
(Pascal et al., 2013), based on Tyler's robust M-estimator (Tyler, 1987) and on
Ledoit and Wolf's shrinkage covariance matrix estimator (Ledoit and Wolf,
2004). These hybrid estimators have the advantage of conveying (i) robustness
to outliers or impulsive samples and (ii) small sample size adequacy to the
classical sample covariance matrix estimator. We consider here the case of
i.i.d. elliptical zero mean samples in the regime where both sample and
population sizes are large. We demonstrate that, under this setting, the
estimators under study asymptotically behave similar to well-understood random
matrix models. This characterization allows us to derive optimal shrinkage
strategies to estimate the population scatter matrix, improving significantly
upon the empirical shrinkage method proposed in (Chen et al., 2011).
