L1-Penalized Quantile Regression in High-Dimensional Sparse Models

Understand

We consider median regression and, more generally, a possibly infinite collection of quantile regressions in high-dimensional sparse models.

• In these models the overall number of regressors $p$ is very large, possibly larger than the sample size $n$, but only $s$ of these regressors have non-zero impact on the conditional quantile of the response variable, where $s$ grows slower than $n$.
• We consider quantile regression penalized by the $\ell_1$-norm of coefficients ($\ell_1$-QR).
• First, we show that $\ell_1$-QR is consistent at the rate $\sqrt{s/n} \sqrt{\log p}$.