Efficient quantile regression for heteroscedastic models

Quantile regression (QR) provides estimates of a range of conditional quantiles. This stands in contrast to traditional regression techniques, which focus on a single conditional mean function. Lee et al. [Regularization of case-specific parameters for robustness and efficiency. Statist Sci. 2012;27(3):350–372] proposed efficient QR by rounding the sharp corner of the loss. The main modification generally involves an asymmetric ℓ2 adjustment of the loss function around zero. We extend the idea of ℓ2 adjusted QR to linear heterogeneous models. The ℓ2 adjustment is constructed to diminish as sample size grows. Conditions to retain consistency properties are also provided.