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

By Jelena Bradic, Mladen Kolar. 2017

28 citations recorded by the source corpus.

Author(s): Bradic, Jelena; Kolar, Mladen | Abstract: 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.

Subfields: Statistics and Probability

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Citations in the corpus

Double/De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers (2018, background): “[2015]; Jankova and Van De Geer [2015, 2016, 2018]; Bradic and Kolar [2017]; Zhu and Bradic [2017, 2018]], which focuses on coefficients in high-dimensional linear and generalized linear regression models, without considering the general linear functionals analyzed here.”
Double/de-biased machine learning using regularized Riesz representers (2018, background): “…et al. (2014); Ning and Liu (2014); Chernozhukov et al. (2015); Neykov et al. (2015); Ren et al. (2015); Jankova and Van De Geer (2015, 2016a,b); Bradic and Kolar (2017); Zhu and Bradic (2017b,a)], which focused on inference on the coefﬁcients in high-dimensional linear and generalized linear…”