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

Lan Wang

Research output: Chapter in Book/Report/Conference proceedingChapter

## Abstract

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) $Q_{Y}(0.5|\mathbf{x})$.

Original language English (US) Handbook of Quantile Regression CRC Press 273-292 20 9781498725293 9781498725286 https://doi.org/10.1201/9781315120256 Published - Jan 1 2017