Robust Procedures in Multivariate Analysis I: Robust Covariance Estimation

By N. Campbell. 1980

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Sign and rank covariance matrices (2000, methodology): “See Maronna (1976), Huber (1977), Campbell (1980) and Kent and Tyler (1991) for M -estimates and Davies (1987), Lopuhaä (1989) and Lopuhaä and Rousseeuw (1991) for so called S-estimates. The minimum volume ellipsoid (MVE) estimator (see Rousseeuw and Leroy (1987)) is given by the smallest regular el”
Robust principal component analysis for computer vision (2001, background): “These reconstruction coefficients can be arbitrarily biased for an outlier.”
A comparative study on diabetes disease diagnosis using neural networks (2009, background): “As soon as data are contaminated, classical PCA breaks down i.e. performance of PCA is seriously influenced by outliers and atypical values (Serneels & Verdonck, 2008; Campbell, 1980).”
ROBPCA: A New Approach to Robust Principal Component Analysis (2005, methodology): “Maronna (1976) and Campbell (1980) proposed using affine-equivariant M-estimators of scatter for this purpose, but these cannot resist many outliers. More recently, Croux and Haesbroeck (2000) used positive-breakdown estimators, such as the minimum covariance determinant (MCD) method (Rousseeuw 1984”
A general methodology for bootstrapping in non-parametric frontier models (2000, methodology): “We use the M-estimator of the covariance matrix described by Campbell (1980). Alternatively, one could use the minimum volume ellipsoid estimator proposed by Rousseeuw (1985) when (p + q) is large, since the breakdown point of M-estimators decreases as the dimensionality increases (see Hampel et al.”
Principal Components Analysis Based on Multivariate MM Estimators With Fast and Robust Bootstrap (2006, methodology): “The latter approach has been proposed and investigated by many authors, starting with Maronna (1976), Campbell (1980), and Devlin, Gnanadesikan, and Kettenring (1981). More recently, Croux and Haesbroeck (2000) compared several robust estimators in a more formal manner.”
Robustness in Data Analysis: Criteria and Methods (2001, background): “Maximum likelihood and M-estimators (Campbell, 1980; Campbell, 1982; Huber, 1981; Lopuhaä, 1992; Maronna, 1976; Meshalkin, 1971; Shurygin, 1994a; Shurygin, 1995; Tyler, 1983; Tyler, 1991), S-estimators (Davies, 1987; Hampel et al.”
Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery (2017, background): “In the statistics literature in 1980s, the predominant approach consists in replacing the standard estimation of the covariance matrix with a robust estimator of the covariance matrix [55], [56].”
High-Breakdown Robust Multivariate Methods (2008, methodology): “Maronna (1976) and Campbell (1980) proposed using affine equivariant M-estimators of scatter for this purpose, but these cannot resist many outliers. Croux and Haesbroeck (2000) used positive-breakdown estimators of scatter such as the MCD and S-estimators. Recently, Salibian-Barrera, Van Aelst and ”
An Object-Oriented Framework for Robust Multivariate Analysis (2009, methodology): “In the earlier works M estimators of location and scatter were used for this purpose (see Devlin et al. 1981; Campbell 1980) but these estimators have the disadvantage of low breakdown point in high dimensions.”
Nearest-Neighbor Variance Estimation (NNVE) (2002, background): “There is a substantial literature on this problem (see, e.g., Campbell 1980; Huber 1981; Lopuhaä 1989; Maronna 1976; Maronna and Yohai 1995; Poston et al. 1997; Rousseeuw and Leroy 1987; Ruppert 1992; Tyler 1994; Woodruff and Rocke 1994).”
Structural equation modeling with heavy tailed distributions (2004, methodology): “Motivated by the unbiasedness of the sample covariance matrix, Campbell (1980) defined another form of M-estimators: ̂ = n∑ i=1 wi xi / n∑ i=1 wi , (11a) ̂ = n∑ i=1 w2i (xi − ̂)(xi − ̂)′ /( n∑ i=1 w2i − 1 ) , (11b) where wi = w{d(xi , ̂, ̂)} for a weight function w(t).”