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Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation. (English) Zbl 1203.62103

Summary: 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 transformations 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.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G99 Nonparametric inference
62H99 Multivariate analysis

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