The Behavior of the Stahel-Donoho Robust Multivariate Estimator

• Published in: Journal of the American Statistical Association Vol. 90; no. 429; pp. 330 - 341
• Main Authors: Maronna, Ricardo A., Yohai, Víctor J.
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis Group 01-03-1995; American Statistical Association; Taylor & Francis Ltd
• Subjects: Bias robustness; Covariance; Covariance matrices; Datasets; Distribution; Estimation bias; Estimators; Exact sciences and technology; Forecasts; Mathematics; Maximum likelihood estimation; Multivariate analysis; Multivariate location scatter; Outliers; Point estimators; Point masses; Probability and statistics; Projection methods; Sciences and techniques of general use; Statistical median; Statistics; Theory and Methods; Weighting functions; Projection method; Outlier; Estimator efficiency; Consistent estimator; Bias; Estimator robustness; Gaussian distribution; Multivariate distribution; Cauchy distribution
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.1995.10476517

The Stahel-Donoho estimators (t, V) of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w(r), where w is a weight function and r is a measure of "outlyingness," obtained by considering all univariate projections of the data. It has a high breakdown point for all dimensions and order √n consistency. The asymptotic bias of V for point mass contamination for suitable weight functions is compared with that of Rousseeuw's minimum volume ellipsoid (MVE) estimator. A simulation shows that for a suitable w, t and V exhibit high efficiency for both normal and Cauchy distributions and are better than their competitors for normal data with point-mass contamination. The performances of the estimators for detecting outliers are compared for both a real and a synthetic data set.