Robustness of the Affine Equivariant Scatter Estimator Based on the Spatial Rank Covariance Matrix

Visuri et al. (2000) proposed a technique for robust covariance matrix estimation based on different notions of multivariate sign and rank. Among them, the spatial rank based covariance matrix estimator that utilizes a robust scale estimator is especially appealing due to its high robustness, computational ease, and good efficiency. Also, it is orthogonally equivariant under any distribution and affinely equivariant under elliptically symmetric distributions. In this paper, we study robustness properties of the estimator with respective to two measures: breakdown point and influence function. More specifically, the upper bound of the finite sample breakdown point can be achieved by a proper choice of univariate robust scale estimator. The influence functions for eigenvalues and eigenvectors of the estimator are derived. They are found to be bounded under some assumptions. Moreover, finite sample efficiency comparisons to popular robust MCD, M, and S estimators are reported.

[1]  H. Oja,et al.  Sign and Rank Covariance Matrices: Statistical Properties and Application to Principal Components Analysis , 2002 .

[2]  Douglas G. Simpson,et al.  Robust Direction Estimation , 1992 .

[3]  R. Serfling,et al.  Spatial Trimming , with Applications to Robustify Sample Spatial Quantile and Outlyingness Functions , and to Construct a New Robust Scatter Estimator , 2010 .

[4]  R. Serfling.  Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation , 2010 .

[5]  Hannu Oja,et al.  INFLUENCE FUNCTION AND ASYMPTOTIC EFFICIENCY OF THE AFFINE EQUIVARIANT RANK COVARIANCE MATRIX , 2004 .

[6]  Hannu Oja,et al.  Estimates of regression coefficients based on the sign covariance matrix , 2002 .

[7]  Hannu Oja,et al.  ON THE EFFICIENCY OF MULTIVARIATE SPATIAL SIGN AND RANK TESTS , 1997 .

[8]  P. Rousseeuw,et al.  Alternatives to the Median Absolute Deviation , 1993 .

[9]  R. Maronna.  Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[10]  Y. Tanaka.  Sensitivity analysis in principal component analysis:influence on the subspace spanned by principal components. , 1988 .

[11]  Hengjian Cui,et al.  Depth weighted scatter estimators , 2005 .

[12]  R. Serfling,et al.  A robust sample spatial outlyingness function , 2013 .

[13]  Hannu Oja,et al.  Affine equivariant multivariate rank methods , 2003 .

[14]  David E. Tyler,et al.  Tests and estimates of shape based on spatial signs and ranks , 2009 .

[15]  Weihua Zhou,et al.  Projection based scatter depth functions and associated scatter estimators , 2010, J. Multivar. Anal..

[16]  Hannu Oja,et al.  k-Step shape estimators based on spatial signs and ranks , 2010 .

[17]  H. Oja.  Multivariate Nonparametric Methods with R , 2010 .

[18]  P. Rousseeuw.  Multivariate estimation with high breakdown point , 1985 .

[19]  David E. Tyler,et al.  Invariant co‐ordinate selection , 2009 .

[20]  David E. Tyler,et al.  On the Breakdown Properties of Some Multivariate M‐Functionals * , 2005 .

[21]  H. Oja,et al.  Sign and rank covariance matrices , 2000 .

[22]  V. Koltchinskii.  M-estimation, convexity and quantiles , 1997 .

[23]  Hannu Oja,et al.  Estimates of Regression Coefficients Based on Lift Rank Covariance Matrix , 2003 .

[24]  Ursula Gather,et al.  A note on Tyler's modification of the MAD for the Stahel-Donoho estimator , 1997 .

[25]  J. Marden.  Some robust estimates of principal components , 1999 .

[26]  C. Croux,et al.  Principal Component Analysis Based on Robust Estimators of the Covariance or Correlation Matrix: Influence Functions and Efficiencies , 2000 .

[27]  David E. Tyler.  Finite Sample Breakdown Points of Projection Based Multivariate Location and Scatter Statistics , 1994 .

[28]  Daniel Gervini,et al.  A robust and efficient adaptive reweighted estimator of multivariate location and scatter , 2003 .

[29]  C. Croux,et al.  The k-step spatial sign covariance matrix , 2010, Adv. Data Anal. Classif..

[30]  David E. Tyler.  A Distribution-Free $M$-Estimator of Multivariate Scatter , 1987 .

[31]  L. Dümbgen.  On Tyler's M-Functional of Scatter in High Dimension , 1998 .

[32]  P. L. Davies,et al.  Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices , 1987 .

[33]  R. Randles,et al.  A practical affine equivariant multivariate median , 2002 .

[34]  Victor J. Yohai,et al.  The Behavior of the Stahel-Donoho Robust Multivariate Estimator , 1995 .

[35]  Christophe Croux,et al.  High breakdown estimators for principal components: the projection-pursuit approach revisited , 2005 .

[36]  Hannu Oja,et al.  Multivariate Nonparametric Tests , 2004 .

[37]  P. L. Davies,et al.  Breakdown and groups , 2005, math/0508497.