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arXiv · 1312.5594

M-Functionals of Multivariate Scatter

Lutz Duembgen·Markus Pauly·Thomas Schweizer
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Abstract

This survey provides a self-contained account of $M$-estimation of multivariate scatter. In particular, we present new proofs for existence of the underlying $M$-functionals and discuss their weak continuity and differentiability. This is done in a rather general framework with matrix-valued random variables. By doing so we reveal a connection between Tyler's (1987) $M$-functional of scatter and the estimation of proportional covariance matrices. Moreover, this general framework allows us to treat a new class of scatter estimators, based on symmetrizations of arbitrary order. Finally these results are applied to $M$-estimation of multivariate location and scatter via multivariate $t$-distributions.

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Lutz Duembgen, Markus Pauly, Thomas Schweizer. 2015-03-19. M-Functionals of Multivariate Scatter. https://doi.org/10.1214/15-ss109

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