On consistency factors and efficiency of robust S-estimators

We tackle the problem of obtaining the consistency factors of robust S-estimators of location and scale both in regression and multivariate analysis. We provide theoretical results, proving new formulae for their calculation and shedding light on the relationship between these factors and important statistical properties of S-estimators. We also give computational advances, suggesting efficient procedures so that hardly any time is lost for their calculation when computing S-estimates. In addition, when the purpose is to fix the efficiency of the scale estimator, we are able to quantify to what extent the approximate algorithms which are currently available provide an acceptable solution, and when it is necessary to resort to the exact formulae. Finally, even if this paper concentrates on S-estimates and Tukey’s Biweight and optimal loss functions, the main results can be easily extended to calculate the tuning consistency factors for other popular loss function and other robust estimators.

[1]  Peter Filzmoser,et al.  An Object-Oriented Framework for Robust Multivariate Analysis , 2009 .

[2]  J. A. Cuesta-Albertos,et al.  Trimming and likelihood: Robust location and dispersion estimation in the elliptical model , 2008, 0811.0503.

[3]  V. Yohai.  HIGH BREAKDOWN-POINT AND HIGH EFFICIENCY ROBUST ESTIMATES FOR REGRESSION , 1987 .

[4]  V. Yohai,et al.  A Fast Algorithm for S-Regression Estimates , 2006 .

[5]  Francesca Torti,et al.  FSDA: A MATLAB toolbox for robust analysis and interactive data exploration , 2012 .

[6]  N. L. Johnson,et al.  Continuous Multivariate Distributions: Models and Applications , 2005 .

[7]  H. P. Lopuhaä.  On the relation between S-estimators and M-estimators of multivariate location and covariance , 1989 .

[8]  A. Atkinson,et al.  Finding an unknown number of multivariate outliers , 2009 .

[9]  P. L. Davies,et al.  The asymptotics of S-estimators in the linear regression model , 1990 .

[10]  G. M. Tallis.  Elliptical and Radial Truncation in Normal Populations , 1963 .

[11]  Domenico Perrotta,et al.  The Forward Search for Very Large Datasets , 2015 .

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

[13]  Kostas Triantafyllopoulos,et al.  On the central moments of the multidimensional Gaussian distribution , 2003 .

[14]  N. Campbell.  Mixture models and atypical values , 1984 .

[15]  Andrea Cerioli,et al.  Multivariate Outlier Detection With High-Breakdown Estimators , 2010 .

[16]  David E. Tyler,et al.  Redescending $M$-Estimates of Multivariate Location and Scatter , 1991 .

[17]  C. Matr'an,et al.  A general trimming approach to robust Cluster Analysis , 2008, 0806.2976.

[18]  On the Optimality of Multivariate S‐Estimators , 2011 .

[19]  Anthony C. Atkinson,et al.  Robust Diagnostic Regression Analysis , 2000 .

[20]  N. L. Johnson,et al.  Continuous Multivariate Distributions, Volume 1: Models and Applications , 2019 .

[21]  Pavel Čížek,et al.  Reweighted least trimmed squares: an alternative to one-step estimators , 2010, TEST.

[22]  Sonja Rieder.  Robust parameter estimation for the Ornstein–Uhlenbeck process , 2012, Stat. Methods Appl..

[23]  Peter J. Rousseeuw,et al.  Robust Regression and Outlier Detection , 2005, Wiley Series in Probability and Statistics.

[24]  L. Isserlis.  ON A FORMULA FOR THE PRODUCT-MOMENT COEFFICIENT OF ANY ORDER OF A NORMAL FREQUENCY DISTRIBUTION IN ANY NUMBER OF VARIABLES , 1918 .

[25]  David E. Tyler.  Robustness and efficiency properties of scatter matrices , 1983 .

[26]  Gert Willems,et al.  Robust and Efficient One-Way MANOVA Tests , 2011 .

[27]  V. Yohai,et al.  Optimal locally robust M-estimates of regression , 1997 .

[28]  Anthony C. Atkinson,et al.  Monitoring robust regression , 2014 .

[29]  C. Croux,et al.  Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator , 1999 .

[30]  W. Härdle,et al.  Robust and Nonlinear Time Series Analysis , 1984 .

[31]  M. Hubert,et al.  High-Breakdown Robust Multivariate Methods , 2008, 0808.0657.

[32]  S. Van Aelst,et al.  Principal Components Analysis Based on Multivariate MM Estimators With Fast and Robust Bootstrap , 2006 .

[33]  Regina Y. Liu,et al.  Multivariate analysis by data depth: descriptive statistics, graphics and inference, (with discussion and a rejoinder by Liu and Singh) , 1999 .

[34]  David M. Rocke.  Robustness properties of S-estimators of multivariate location and shape in high dimension , 1996 .

[35]  Björn Holmquist,et al.  Moments and cumulants of the multivariate normal distribution. , 1988 .

[36]  Christophe Croux,et al.  Robust M-Estimation of Multivariate GARCH models , 2010, Comput. Stat. Data Anal..

[37]  Luis Angel García-Escudero,et al.  Generalized Radius Processes for Elliptically Contoured Distributions , 2005 .

[38]  Peter J. Rousseeuw,et al.  ROBUST REGRESSION BY MEANS OF S-ESTIMATORS , 1984 .

[39]  THE TEACHING OF ROBUST STATISTICS FOR REGRESSION , 2006 .

[40]  Extending the Hausman Test to Check for the Presence of Outliers , 2011 .

[41]  M. Hubert,et al.  A Deterministic Algorithm for Robust Location and Scatter , 2012 .

[42]  Christophe Croux,et al.  Testing the information matrix equality with robust estimators , 2006 .

[43]  Ola Hössjer,et al.  On the optimality of S-estimators☆ , 1992 .