Robust estimation for the multivariate linear model based on a τ -scale

We introduce a class of robust estimates for multivariate linear models. The regression coefficients and the covariance matrix of the errors are estimated simultaneously by minimizing the determinant of the covariance matrix estimate, subject to a constraint on a robust scale of the Mahalanobis norm...

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Bibliographic Details
Published in:Journal of multivariate analysis 2006-08, Vol.97 (7), p.1600-1622
Main Authors: Ben, Marta García, Martínez, Elena, Yohai, Víctor J.
Format: Article
Language:English
Subjects:
[formula omitted]-estimates
Exact sciences and technology
Linear inference, regression
Mathematics
Multivariate analysis
Multivariate regression
Parametric inference
Probability and statistics
Robust estimation
Sciences and techniques of general use
Statistics
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Summary:We introduce a class of robust estimates for multivariate linear models. The regression coefficients and the covariance matrix of the errors are estimated simultaneously by minimizing the determinant of the covariance matrix estimate, subject to a constraint on a robust scale of the Mahalanobis norms of the residuals. By choosing a τ -estimate as a robust scale, the resulting estimates combine good robustness properties and asymptotic efficiency under Gaussian errors. These estimates are asymptotically normal and in the case where the errors have an elliptical distribution, their asymptotic covariance matrix differs only by a scalar factor from the one corresponding to the maximum likelihood estimate. We derive the influence curve and prove that the breakdown point is close to 0.5. A Monte Carlo study shows that our estimates compare favorably with respect to S-estimates.
ISSN:0047-259X
1095-7243
DOI:10.1016/j.jmva.2005.08.007