Depth‐weighted robust multivariate regression with application to sparse data

Abstract

A robust method for multivariate regression is developed based on robust estimators of the joint location and scatter matrix of the explanatory and response variables using the notion of data depth. The multivariate regression estimator possesses desirable affine equivariance properties, achieves the best breakdown point of any affine equivariant estimator, and has an influence function which is bounded in both the response as well as the predictor variable. To increase the efficiency of this estimator, a re‐weighted estimator based on robust Mahalanobis distances of the residual vectors is proposed. In practice, the method is more stable than existing methods that are constructed using subsamples of the data. The resulting multivariate regression technique is computationally feasible, and turns out to perform better than several popular robust multivariate regression methods when applied to various simulated data as well as a real benchmark data set. When the data dimension is quite high compared to the sample size it is still possible to use meaningful notions of data depth along with the corresponding depth values to construct a robust estimator in a sparse setting. The Canadian Journal of Statistics 45: 164–184; 2017 © 2017 Statistical Society of Canada

Envelopes for elliptical multivariate linear regression

“…Bias correction for the maximum likelihood estimator and adjustments of the likelihood-ratio statistics are also derived for this general model (see [38], [37]). The elliptical distributions can also be used as the basis to consider robustness in multivariate linear regression, as in [14], [21], [29], [36], [42], [50] and many others. Nevertheless the envelope model under the context of elliptical multivariate regression has not yet been implemented.…”