M-Functionals of Multivariate Scatter

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.

References in corpus (1)

• New Algorithms for $M$-Estimation of Multivariate Scatter and Location

Cited by in corpus (10)

• Robust Sparse Covariance Estimation by Thresholding Tyler's M-Estimator
• New Algorithms for $M$-Estimation of Multivariate Scatter and Location
• Geodesic Convexity and Regularized Scatter Estimators
• A universal framework for learning the elliptical mixture model
• Non-Gaussian component analysis: testing the dimension of the signal subspace
• Approximating Symmetrized Estimators of Scatter via Balanced Incomplete U-Statistics
• On characterizations of the covariance matrix
• Mean field repulsive Kuramoto models: Phase locking and spatial signs
• Gini Covariance Matrix and its Affine Equivariant Version
• A method for sparse and robust independent component analysis