Approximate nonparametric quantile regression in reproducing kernel Hilbert spaces via random projection

Nonparametric quantile regression is a commonly used nonlinear quantile model. One general and popular approach is based on the use of kernels within a reproducing kernel Hilbert space (RKHS) framework, with the smoothing splines estimation as a special case. However, when the sample size n is large, the computational burden is heavy. Motivated by the recent advances in random projection for kernel nonparametric (mean) ridge regression (KRR), we consider an m-dimensional random projection approach for kernel quantile regression (KQR) with m << n. We establish a theoretical result showing that the sketched KQR still achieves the minimax convergence rate when m is at least as large as the effective statistical dimension of the problem. Some Monte Carlo studies are carried out for illustration purposes. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

Nonparametric quantile regression is commonly used for nonlinear quantile modeling, with a kernel approach in a reproducing kernel Hilbert space framework. To address heavy computational burden with large sample sizes, a random projection approach with m << n is considered. Theoretical results show that sketched KQR achieves minimax convergence rate when m is at least as large as the effective statistical dimension of the problem.