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.

Large sample and robust properties of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmqr1ngBPrgitL % xBI9gBaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B % 1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbfv3ySLgzGueE0j % xyaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFfea % 0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0- % vqpWqaaeaabiGaciaadaqabeaaeaqbayaaaOqaaiqbdYeamzaaiaWa % aWbaaSqabeaacqaIYaGmaaaaaa!43D3! $$ \tilde L^2 $$ -median

“…However, it is easy to extend most of the results by using weak covariance functionals (see definitions in [30]) of F instead of Σ . Zuo proved that L 2 -depth was a statistical depth function in the sense of Definition 2.1 in [9].…”

“…In this paper, the depth and the median derived by the latter method are called L 2 -depth and L 2 -median, respectively. As pointed out by an anonymous referee, in the latest paper of Serfling [30] , TR method and other standardization ways are discussed extensively. Some of the results are very stimulating for our paper.…”

“…Some of the results are very stimulating for our paper. In particular, the Mahalanobis spatial median defined in [30] may be regarded as the weaker version of L 2 -median. However, except Isogai [12] , there is not much discussion about L 2 -median.…”

Discussion of “Multivariate Functional Outlier Detection”, by Mia Hubert, Peter Rousseeuw and Pieter Segaert

“…In Serfling (2006) and Serfling (2010), a Depth-Outlyingness-Quantile-Rank (D-O-Q-R) paradigm is introduced and discussed. Under this paradigm, depth, outlyingness, quantile, and rank functions are seen as strictly equivalent objects; in particular, depth and outlyingness are inversely linked through, e.g., relations of the form D = 1/(1 + O) or D = 1 − O; see also Serfling and Zuo (2010).…”

From Depth to Local Depth: A Focus on Centrality

“…Indeed, not only does depth provide a robust multivariate location functional (through the deepest point), but it also yields information about spread, shape, and symmetry (through depth regions; Serfling 2004) and even characterizes the underlying distribution under very mild conditions (see Kong and Zuo 2010 and the references therein). Celebrated instances of such depths include Tukey's halfspace depth (Tukey 1975), Liu's simplicial depth (Liu 1990), the projection depth (Zuo 2003), or the Mahalanobis depth (see, e.g., Zuo and Serfling 2000a).…”