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Hypothesis tests in models whose dimension far exceeds the sample size can beformulated much like the classical studentized tests only after the initialbias of estimation is removed successfully. The theory of debiased estimatorscan be developed in the context of quantile regression models for a fixedquantile value. However, it is frequently desirable to formulate tests based onthe quantile regression process, as this leads to more robust tests and morestable confidence sets. Additionally, inference in quantile regression requiresestimation of the so called sparsity function, which depends on the unknowndensity of the error. In this paper we consider a debiasing approach for theuniform testing problem. We develop high-dimensional regression rank scores andshow how to use them to estimate the sparsity function, as well as how to adaptthem for inference involving the quantile regression process. Furthermore, wedevelop a Kolmogorov-Smirnov test in a location-shift high-dimensional modelsand confidence sets that are uniformly valid for many quantile values. The maintechnical result are the development of a Bahadur representation of thedebiasing estimator that is uniform over a range of quantiles and uniformconvergence of the quantile process to the Brownian bridge process, which areof independent interest. Simulation studies illustrate finite sample propertiesof our procedure.

Uniform Inference for High-dimensional Quantile Regression: Linear Functionals and Regression Rank Scores