  Hypothesis tests in models whose dimension far exceeds the sample size can be
formulated much like the classical studentized tests only after the initial
bias of estimation is removed successfully. The theory of debiased estimators
can be developed in the context of quantile regression models for a fixed
quantile value. However, it is frequently desirable to formulate tests based on
the quantile regression process, as this leads to more robust tests and more
stable confidence sets. Additionally, inference in quantile regression requires
estimation of the so called sparsity function, which depends on the unknown
density of the error. In this paper we consider a debiasing approach for the
uniform testing problem. We develop high-dimensional regression rank scores and
show how to use them to estimate the sparsity function, as well as how to adapt
them for inference involving the quantile regression process. Furthermore, we
develop a Kolmogorov-Smirnov test in a location-shift high-dimensional models
and confidence sets that are uniformly valid for many quantile values. The main
technical result are the development of a Bahadur representation of the
debiasing estimator that is uniform over a range of quantiles and uniform
convergence of the quantile process to the Brownian bridge process, which are
of independent interest. Simulation studies illustrate finite sample properties
of our procedure.
