The focus is on robust regression methods for problems where the predictor matrix has full rank and where it is rank deficient. For the first situation, various robust regression methods have been introduced, and here an overview of the most important proposals is given. For the latter case, robust partial least squares regression is discussed. The way of downweighting outlying observations is important. Using continuous weights (leading to "soft" robust methods) has advantages over 0/1 weights in terms of statistical efficiency of the estimators. This will be illustrated for both types of regression problems. Soft methods are particularly useful in high-dimensional settings.
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