Statistics Surveys

$M$-functionals of multivariate scatter

Lutz Dümbgen, Markus Pauly, and Thomas Schweizer

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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 (1987a) $M$-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.

Article information

Statist. Surv., Volume 9 (2015), 32-105.

First available in Project Euclid: 20 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties 62G35: Robustness 62H12: Estimation 62H99: None of the above, but in this section

Coercivity convexity matrix exponential function multivariate $t$-distribution scatter functionals weak continuity weak differentiablity


Dümbgen, Lutz; Pauly, Markus; Schweizer, Thomas. $M$-functionals of multivariate scatter. Statist. Surv. 9 (2015), 32--105. doi:10.1214/15-SS109.

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