High breakdown estimators for principal components: the projection-pursuit approach revisited.

Li and Chen (J. Amer. Statist. Assoc. 80 (1985) 759) proposed a method for principal components using projection-pursuit techniques. In classical principal components one searches for directions with maximal variance, and their approach consists of replacing this variance by a robust scale measure. Li and Chen showed that this estimator is consistent, qualitative robust and inherits the breakdown point of the robust scale estimator. We complete their study by deriving the influence function of the estimators for the eigenvectors, eigenvalues and the associated dispersion matrix. Corresponding Gaussian efficiencies are presented as well. Asymptotic normality of the estimators has been treated in a paper of Cui et al. (Biometrika 90 (2003) 953), complementing the results of this paper. Furthermore, a simple explicit version of the projection-pursuit based estimator is proposed and shown to be fast to compute, orthogonally equivariant, and having the maximal finite-sample breakdown point property. We will illustrate the method with a real data example. (c) 2004 Elsevier Inc. All rights reserved.breakdown point; dispersion matrix; influence function; principal components analysis; projection-pursuit; robustness; dispersion matrices; s-estimators; robust; covariance; location; scale;

Fast and robust estimation of the multivariate errors in variables model.

In the multivariate errors in variable models one wishes to retrieve a linear relationship of the form y = ß x + a, where both x and y can be multivariate. The variables y and x are not directly measurable, but observed with measurement error. The classical approach to estimate the multivariate errors in variable model is based on an eigenvector analysis of the joint covariance matrix of the observations. In this paper a projection-pursuit approach is proposed to estimate the unknown parameters. Focus is on projection indices based on half-samples. These will lead to robust estimators, which can be computed using fast algorithms. Consistency of the procedure is shown, without needing to make distributional assumptions on the x-variables. A simulation study gives insight in the robustness and the efficiency of the procedure.Algorithms; Consistency; Covariance; Efficiency; Errors in variables; Estimator; Matrix; Measurement; Model; Models; Multivariate statistics; Principal component analysis; Projection-pursuit; Robust estimation; Robustness; Simulation; Studies; Variables;

High breakdown estimators for principal components: the projection-pursuit approach revisited

Li and Chen (J. Amer. Statist. Assoc. 80 (1985) 759) proposed a method for principal components using projection-pursuit techniques. In classical principal components one searches for directions with maximal variance, and their approach consists of replacing this variance by a robust scale measure. Li and Chen showed that this estimator is consistent, qualitative robust and inherits the breakdown point of the robust scale estimator. We complete their study by deriving the influence function of the estimators for the eigenvectors, eigenvalues and the associated dispersion matrix. Corresponding Gaussian efficiencies are presented as well. Asymptotic normality of the estimators has been treated in a paper of Cui et al. (Biometrika 90 (2003) 953), complementing the results of this paper. Furthermore, a simple explicit version of the projection-pursuit based estimator is proposed and shown to be fast to compute, orthogonally equivariant, and having the maximal finite-sample breakdown point property. We will illustrate the method with a real data example. (c) 2004 Elsevier Inc. All rights reserved.status: publishe

Robust estimation in the simple errors-in-variables model

Fekri and Ruiz-Gazen [2004. Robust weighted orthogonal regression in the errors-in-variables model. J. Multivar. Anal. 88, 89-108] propose a new class of robust estimators in the errors-in-variables linear model (EIV model) under the assumption that the measurement error variances are equal. In the present paper, we extend this new class of estimators to other usual error variances assumptions for the simple EIV model and calculate their influence functions and their asymptotic distributions in the elliptical case.Errors-in-variables model Orthogonal regression B-robustness Influence function M-estimators S-estimators MCD estimator

Robust weighted orthogonal regression in the errors-in-variables model

This paper focuses on robust estimation in the structural errors-in-variables (EV) model. A new class of robust estimators, called weighted orthogonal regression estimators, is introduced. Robust estimators of the parameters of the EV model are simply derived from robust estimators of multivariate location and scatter such as the M-estimators, the S-estimators and the MCD estimator. The influence functions of the proposed estimators are calculated and shown to be bounded. Moreover, we derive the asymptotic distributions of the estimators and illustrate the results on simulated examples and on a real-data set.Errors-in-variables model General least squares Robustness Influence function M-estimators S-estimators MCD estimator

Fast and robust estimation of the multivariate errors in variables model

In the multivariate errors in variable models one wishes to retrieve a linear relationship of the form y = ß x + a, where both x and y can be multivariate. The variables y and x are not directly measurable, but observed with measurement error. The classical approach to estimate the multivariate errors in variable model is based on an eigenvector analysis of the joint covariance matrix of the observations. In this paper a projection-pursuit approach is proposed to estimate the unknown parameters. Focus is on projection indices based on half-samples. These will lead to robust estimators, which can be computed using fast algorithms. Consistency of the procedure is shown, without needing to make distributional assumptions on the x-variables. A simulation study gives insight in the robustness and the efficiency of the procedure.status: publishe