Oracle Estimation of a Change Point in High Dimensional Quantile Regression

TL;DR

This paper introduces a novel high-dimensional quantile regression method that automatically detects and estimates a change point without prior knowledge, achieving oracle properties and applicable to general M-estimation frameworks.

Contribution

It develops $ au$-penalized estimators for both regression coefficients and change points, with proven oracle properties and no need for perfect covariate selection.

Findings

Estimator achieves oracle property in asymptotic distribution.

Method effectively discriminates between homogeneous and change point models.

Validated through Monte Carlo simulations and real data application.

Abstract

In this paper, we consider a high-dimensional quantile regression model where the sparsity structure may differ between two sub-populations. We develop $\ell_1$-penalized estimators of both regression coefficients and the threshold parameter. Our penalized estimators not only select covariates but also discriminate between a model with homogeneous sparsity and a model with a change point. As a result, it is not necessary to know or pretest whether the change point is present, or where it occurs. Our estimator of the change point achieves an oracle property in the sense that its asymptotic distribution is the same as if the unknown active sets of regression coefficients were known. Importantly, we establish this oracle property without a perfect covariate selection, thereby avoiding the need for the minimum level condition on the signals of active covariates. Dealing with high-dimensional quantile regression with an unknown change point calls for a new proof technique since the quantile loss function is non-smooth and furthermore the corresponding objective function is non-convex with respect to the change point. The technique developed in this paper is applicable to a general M-estimation framework with a change point, which may be of independent interest. The proposed methods are then illustrated via Monte Carlo experiments and an application to tipping in the dynamics of racial segregation.