Model selection in high-dimensional quantile regression with seamless $L_0$ penalty

TL;DR

This paper introduces a seamless $L_0$ quantile estimator for high-dimensional linear models, providing convergence rates, variable selection consistency, asymptotic normality, and a BIC criterion for tuning parameter selection.

Contribution

It proposes a novel seamless $L_0$ quantile estimator that handles high-dimensional data without moment assumptions, with proven convergence, variable selection consistency, and asymptotic normality.

Findings

Estimator converges at a specified rate.

Correctly identifies zero and nonzero parameters.

Provides asymptotic normality for nonzero parameter estimators.

Abstract

In this paper we are interested in parameters estimation of linear model when number of parameters increases with sample size. Without any assumption about moments of the model error, we propose and study the seamless $L_0$ quantile estimator. For this estimator we first give the convergence rate. Afterwards, we prove that it correctly distinguishes between zero and nonzero parameters and that the estimators of the nonzero parameters are asymptotically normal. A consistent BIC criterion to select the tuning parameters is given.