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A lack-of-fit test for quantile regression. (English) Zbl 1043.62039
Summary: We propose an omnibus lack-of-fit test for linear or nonlinear quantile regression based on a cusum process of the gradient vector. The test does not involve nonparametric smoothing but is consistent for all nonparametric alternatives without any moment conditions on the regression error. In addition, the test is suitable for detecting the local alternatives of any order arbitrarily close to \(n^{-1/2}\) from the null hypothesis. The limiting distribution of the proposed test statistic is non-Gaussian but can be characterized by a Gaussian process. We propose a simple sequential resampling scheme to carry out the test whose nominal levels are well approximated in our empirical study for

MSC:
62G10 Nonparametric hypothesis testing
62G08 Nonparametric regression and quantile regression
62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
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