Robust subgroup-classifier learning and testing in change-plane regressions

Considered here are robust subgroup-classifier learning and testing in change-plane regressions with heavy-tailed errors, which can identify subgroups as a basis for making optimal recommendations for individualized treatment. A new subgroup classifier is proposed by smoothing the indicator function, which is learned by minimizing the smoothed Huber loss. Nonasymptotic properties and the Bahadur representation of estimators are established, in which the proposed estimators of the grouping difference parameter and baseline parameter achieve sub-Gaussian tails. The hypothesis test considered here belongs to the class of test problems for which some parameters are not identifiable under the null hypothesis. The classic supremum of the squared score test statistic may lose power in practice when the dimension of the grouping parameter is large, so to overcome this drawback and make full use of the data's heavy-tailed error distribution, a robust weighted average of the squared score test statistic is proposed, which achieves a closed form when an appropriate weight is chosen. Asymptotic distributions of the proposed robust test statistic are derived under the null and alternative hypotheses. The proposed robust subgroup classifier and test statistic perform well on finite samples, and their performances are shown further by applying them to a medical dataset. The proposed procedure leads to the immediate application of recommending optimal individualized treatments.