Nonconvex penalized quantile regression: A review of methods, theory and algorithms

Quantile regression is now a widely recognized useful alternative to the classical least-squares regression. It was introduced in the seminal paper of Koenker and Bassett (1978b). Given a response variable Y and a vector of covariates x $ \mathbf{x} $, quantile regression estimates the effects of x $ \mathbf{x} $ on the conditional quantile of Y. Formally, τ the $ \tau $ th (0 < τ < 1 $ 0<\tau <1 $) conditional quantile of Y given x $ \mathbf{x} $ is defined as Q Y (τ | x) = inf {t : F Y | x (t) ≥ τ } $ Q_{Y}(\tau |\mathbf{x}) = \inf \{t:F_{Y|\mathbf{x}}(t) \ge \tau \} $, where F Y | x $ F_{Y|\mathbf{x}} $ is the 274conditional cumulative distribution function of Y given x $ \mathbf{x} $. An important special case of quantile regression is the least absolute deviation (LAD) regression (Koenker and Bassett, 1978a), which estimates the conditional median Q Y (0.5 | x) $ Q_{Y}(0.5|\mathbf{x}) $.