# Nonconvex Penalized Quantile Regression: A Review of Methods, Theory and Algorithms

Authored by: Lan Wang

# Handbook of Quantile Regression

Print publication date:  October  2017
Online publication date:  October  2017

Print ISBN: 9781498725286
eBook ISBN: 9781315120256

10.1201/9781315120256-16

#### 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, the τ $\tau$ 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 conditional 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 ) $Q_{Y}(0.5|\mathbf{x})$ .

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