  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.
