Significance testing in quantile regression

TL;DR

This paper introduces a new significance test for predictors in multivariate nonparametric quantile regression, demonstrating its ability to detect subtle local alternatives and validating its performance through simulations.

Contribution

It proposes a novel stochastic process-based test that converges to a Gaussian process and can detect local alternatives at any rate, improving upon existing methods.

Findings

Test converges to a Gaussian process under the null hypothesis.

Capable of detecting local alternatives with rate a_n where a_n√n→∞.

Simulation study shows good finite sample properties of the bootstrap test.

Abstract

We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that - in contrast to the nonparametric approach based on estimation of $L^2$-distances - the new test is able to detect local alternatives which converge to the null hypothesis with any rate $a_n \to 0$ such that $a_n \sqrt{n} \to \infty$ (here $n$ denotes the sample size). We also present a small simulation study illustrating the finite sample properties of a bootstrap version of the the corresponding Kolmogorov-Smirnov test.