Robust regression against heavy heterogeneous contamination

The $$\gamma $$ γ -divergence is well-known for having strong robustness against heavy contamination. By virtue of this property, many applications via the $$\gamma $$ γ -divergence have been proposed. There are two types of $$\gamma $$ γ -divergence for the regression problem, in which the base measures are handled differently. In this study, these two $$\gamma $$ γ -divergences are compared, and a large difference is found between them under heterogeneous contamination, where the outlier ratio depends on the explanatory variable. One $$\gamma $$ γ -divergence has the strong robustness even under heterogeneous contamination. The other does not have in general; however, it has under homogeneous contamination, where the outlier ratio does not depend on the explanatory variable, or when the parametric model of the response variable belongs to a location-scale family in which the scale does not depend on the explanatory variables. Hung et al. (Biometrics 74(1):145–154, 2018) discussed the strong robustness in a logistic regression model with an additional assumption that the tuning parameter $$\gamma $$ γ is sufficiently large. The results obtained in this study hold for any parametric model without such an additional assumption.