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This article studies two regularized robust estimators of scatter matricesproposed (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 onLedoit and Wolf's shrinkage covariance matrix estimator (Ledoit and Wolf,2004). These hybrid estimators have the advantage of conveying (i) robustnessto outliers or impulsive samples and (ii) small sample size adequacy to theclassical sample covariance matrix estimator. We consider here the case ofi.i.d. elliptical zero mean samples in the regime where both sample andpopulation sizes are large. We demonstrate that, under this setting, theestimators under study asymptotically behave similar to well-understood randommatrix models. This characterization allows us to derive optimal shrinkagestrategies to estimate the population scatter matrix, improving significantlyupon the empirical shrinkage method proposed in (Chen et al., 2011).

Large Dimensional Analysis and Optimization of Robust Shrinkage Covariance Matrix Estimators