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Significance testing in quantile regression. (English) Zbl 1337.62084
Summary: 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.

MSC:
62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
62G30 Order statistics; empirical distribution functions
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