Modified check loss for efficient estimation via model selection in quantile regression

The check loss function is used to define quantile regression. In cross-validation, it is also employed as a validation function when the true distribution is unknown. However, our empirical study indicates that validation with the check loss often leads to overfitting the data. In this work, we suggest a modified or L2-adjusted check loss which rounds the sharp corner in the middle of check loss. This has the effect of guarding against overfitting to some extent. The adjustment is devised to shrink to zero as sample size grows. Through various simulation settings of linear and nonlinear regressions, the improvement due to modification of the check loss by quadratic adjustment is examined empirically.