Robust regression under asymmetric or/and non-constant variance error by simultaneously training conditional quantiles

We consider regression problems under asymmetric or/and non-constant variance error. We see this problem in several fields such as insurance premium estimation, medical cost analysis, etc. Applying the method of Least Squares (LS) to this problem yields unstable solution because of outliers that appears on one side of regression surfaces. Conventional robust techniques to deal with outliers, which intend to discard or down-weight the outliers equally from both sides of regression surfaces, does not help for asymmetric error. In this paper, we propose an robust regression estimator (an estimator of the conditional mean) under asymmetric or/and non-constant variance error by simultaneously training conditional quantiles in multi-layer perceptron (MLP). This is considered as a kind of learning from hint or multitask learning approach, i.e., we train the conditional quantile estimator as hints or extra tasks to improve generalization properties of the conditional mean estimator. Numerical experiments and an application to medical cost estimation problem have shown that our proposal has robustness and good generalization properties.