On the Breakdown Properties of Some Multivariate M‐Functionals *

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A canonical definition of shape (2008, unknown): “…and Randles (2002), and Randles (2000)), (ii) S(Σ) = (trΣ)/k (Dümbgen (1998), Ollila et al. (2004), and Tyler (1987)), and (iii) S(Σ) = |Σ|1/k (Dümbgen and Tyler (2005), Hallin and Paindaveine (2008a), SalibianBarrera et al. (2006), Taskinen et al. (2006), and Tatsuoka and Tyler…”
Symmetrised M-estimators of multivariate scatter (2007, methodology): “The breakdown properties of Dümbgen’s estimator have been studied by Dümbgen and Tyler [4] and the breakdown point was found to be 1−√1− 1/k in case of special kind of contamination.”
Invariant co‐ordinate selection (2009, methodology): “p+1/, such as the M-estimate corresponding to the maximum likelihood estimate for an elliptical Cauchy distribution (Dümbgen and Tyler, 2005), together with a corresponding one-step W -estimate for which ψ.s/ = s u2.s/ → 0 as s → ∞. Such a one-step W -estimate of scatter has a redescending influence”
Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation (2010, background): “(14) WC functionals are known in the literature as ‘scatter functionals’ (e.g. Dümbgen and Tyler 2005) and ‘shape functionals’ (e.g. Paindaveine 2008; TCDO 2009), among other labels.”