  The cellwise robust M regression estimator is introduced as the first
estimator of its kind that intrinsically yields both a map of cellwise outliers
consistent with the linear model, and a vector of regression coefficients that
is robust against vertical outliers and leverage points. As a by-product, the
method yields a weighted and imputed data set that contains estimates of what
the values in cellwise outliers would need to amount to if they had fit the
model. The method is illustrated to be equally robust as its casewise
counterpart, MM regression. The cellwise regression method discards less
information than any casewise robust estimator. Therefore, predictive power can
be expected to be at least as good as casewise alternatives. These results are
corroborated in a simulation study. Moreover, while the simulations show that
predictive performance is at least on par with casewise methods if not better,
an application to a data set consisting of compositions of Swiss nutrients,
shows that in individual cases, CRM can achieve a significantly higher
predictive accuracy compared to MM regression.
