Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators

Romain Couillet (1, 2) , Matthew Mckay (3)

1 Chaire Radio Flexible - Chaire Radio Flexible Alcatel-Lucent/Supélec
2 E3S - Supélec Sciences des Systèmes
3 HKU - The University of Hong Kong

This article studies two regularized robust estimators of scatter matrices pro-posed 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, improv-ing significantly upon the empirical shrinkage method proposed in (Chen et al., 2011).

Mots clés: random matrix theory; robust estimation; linear shrinkage

Domaines: Informatique [cs]; Mathématiques [math]; Sciences de l'ingénieur [physics]

Romain Couillet, Matthew Mckay. Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators. Journal of Multivariate Analysis, 2014, 131, pp.99 - 120. ⟨10.1016/j.jmva.2014.06.018⟩. ⟨hal-01098854⟩