The asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions

TL;DR

This paper demonstrates that the spatial sign covariance matrix (SSCM) is asymptotically inadmissible compared to affine equivariant estimators like Tyler's matrix, especially impacting principal component analysis.

Contribution

It provides a detailed analysis showing the inefficiency of SSCM relative to Tyler's scatter matrix for elliptically symmetric distributions, both asymptotically and in finite samples.

Findings

SSCM is asymptotically inadmissible with a larger variance-covariance matrix.

The inefficiency of SSCM is most pronounced in principal component analysis.

Finite sample simulations confirm the asymptotic inefficiency results.

Abstract

The asymptotic efficiency of the spatial sign covariance matrix (SSCM) relative to affine equivariant estimates of scatter is studied in detail. In particular, the SSCM is shown to be asymptoticaly inadmissible, i.e. the asymptotic variance-covariance matrix of the consistency corrected SSCM is uniformly smaller than that of its affine equivariant counterpart, namely Tyler's scatter matrix. Although the SSCM has often been recommended when one is interested in principal components analysis, the degree of the inefficiency of the SSCM is shown to be most severe in situations where principal components are of most interest. A finite sample simulation shows the inefficiency of the SSCM also holds for small sample sizes, and that the asymptotic relative efficiency is a good approximation to the finite sample efficiency for relatively modest sample sizes.