Uniform Inference for High-dimensional Quantile Regression: Linear Functionals and Regression Rank Scores
This work addresses the need for robust and stable hypothesis tests in high-dimensional quantile regression models, which is important for statisticians and data analysts dealing with large datasets, though it appears incremental as it builds on existing debiasing methods.
The paper tackles the problem of uniform inference for high-dimensional quantile regression by developing a debiasing approach and regression rank scores to estimate the sparsity function and adapt for quantile process inference, resulting in a Kolmogorov-Smirnov test and confidence sets that are uniformly valid across multiple quantiles, with simulation studies demonstrating finite sample properties.
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.