Minimax bias-robust estimation of the dispersion matrix of a multivariate distribution

The Annals of Statistics · 1998 · Volume 26 · Issue 6

Abstract

Maronna defines affine equivariant $M$-estimators for multivariate location and scatter. They are particularly suited for estimating the pseudo-covariance or scatter matrix of an elliptical population. By defining the bias of a dispersion matrix properly, we consider the maximum bias of an $M$-estimator over an $\varepsilon$ -neighborhood of the underlying elliptical distribution (location known). We find that Tyler’s estimator minimizes the maximum bias.

Keywords

$M$-estimation, 62G05, 62H10, 62H12, bias, Covariance matrix, elliptical distribution, mini-max estimation, multivariate scatter, pseudocovariance matrix, robustness

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