Abstract
ABSTRACTIn this article, we consider a high-dimensional quantile regression model where the sparsity structure may differ between two sub-populations. We develop ℓ1-penalized estimators of both regression coefficients and the threshold parameter. Our penalized estimators not only select covariates but also discriminate between a model with homogenous 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 nonsmooth and furthermore the corresponding objective function is nonconvex with respect to the change point. The technique developed in this article 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. Supplementary materials for this article are available online.
Highlights
In this article, we consider a high-dimensional quantile regression model where the sparsity structure may differ between two sub-populations, thereby allowing for a possible change point in the model
We investigate the existence of tipping in the dynamics of racial segregation using the dataset
We have shown among other things that our estimator of the change point achieves an oracle property without relying on a perfect covariate selection, thereby avoiding the need for the minimum level condition on the signals of active covariates
Summary
We consider a high-dimensional quantile regression model where the sparsity structure (e.g., identities and effects of contributing regressors) may differ between two sub-populations, thereby allowing for a possible change point in the model. There are several important differences: first, Lee, Seo, and Shin (2016) considered a high-dimensional mean regression model with a homoscedastic normal error and with deterministic covariates; second, their method consists of one-step least-square estimation with an 1 penalty; third, they derive nonasymptotic oracle inequalities similar to those in Bickel, Ritov, and Tsybakov (2009) but do not provide any distributional result on the estimator of the change point. We allow for heteroscedastic and nonnormal regression errors and stochastic covariates These changes coupled with the fact that the quantile regression objective function is nonconvex with respect to the threshold parameter τ0 raise new challenges.
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