Composite quantile regression and the oracle Model Selection Theory

TL;DR

This paper introduces composite quantile regression (CQR), a new method for linear regression that remains effective even with infinite error variance, offering superior efficiency over traditional least squares methods.

Contribution

The paper develops CQR, extending oracle model selection theory to cases with infinite error variance and demonstrating its efficiency advantages over least squares.

Findings

CQR maintains oracle properties with infinite error variance.

CQR is over 70% more efficient than least squares.

CQR can be arbitrarily more efficient than least squares.

Abstract

Coefficient estimation and variable selection in multiple linear regression is routinely done in the (penalized) least squares (LS) framework. The concept of model selection oracle introduced by Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] characterizes the optimal behavior of a model selection procedure. However, the least-squares oracle theory breaks down if the error variance is infinite. In the current paper we propose a new regression method called composite quantile regression (CQR). We show that the oracle model selection theory using the CQR oracle works beautifully even when the error variance is infinite. We develop a new oracular procedure to achieve the optimal properties of the CQR oracle. When the error variance is finite, CQR still enjoys great advantages in terms of estimation efficiency. We show that the relative efficiency of CQR compared to the least squares is greater than 70% regardless the error distribution. Moreover, CQR could be much more efficient and sometimes arbitrarily more efficient than the least squares. The same conclusions hold when comparing a CQR-oracular estimator with a LS-oracular estimator.