Significance testing in quantile regression

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 corresponding Kolmogorov-Smirnov test.