Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation

Abstract

Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardisation of multivariate data to produce equivariance or invariance of procedures, three important types of matrix-valued functional are studied: ‘weak covariance’ (or ‘shape’), ‘transformation–retransformation’ (TR), and ‘strong invariant coordinate system’ (SICS). The clarification of TR affine equivariant versions of the sample spatial quantile function is obtained. It is seen that geometric artefacts of SICS-standardised data are invariant under affine transformation of the original data followed by standardisation using the same SICS functional, subject only to translation and homogeneous scale change. Some applications of SICS standardisation are described.

Keywords:

• multivariate quantile functions
• multivariate outlyingness functions
• affine equivariance and invariance
• covariance functionals
• transformation–retransformation functionals
• invariant coordinate system functionals
• nonparametric methods

AMS 2000 Subject Classification :

• Primary: 62G99
• Secondary: 62H99

Acknowledgements

The author greatly appreciates several stimulating discussions with Marc Hallin and Davy Paindaveine, useful input from Hannu Oja, and constructive comments by three anonymous reviewers. This has led to substantial improvements in the manuscript. The author also thanks Satyaki Mazumder for finding errors in earlier drafts. The support of National Science Foundation Grants DMS-0103698 and DMS-0805786 and National Security Agency Grant H98230-08-1-0106 is gratefully acknowledged.