Abstract: In this paper the nonparametric quantile regression model is considered in a location-scale context. The asymptotic properties of the empirical independence process based on covariates and estimated residuals are investigated. In particular an asymptotic expansion and weak convergence to a Gaussian process are proved. The results can be applied to test for validity of the location-scale model, and they allow one to derive various specification tests in conditional quantile location-scale models. A test for monotonicity of the conditional quantile curve is investigated. For the test for validity of the location-scale model, as well as for the monotonicity test, smooth residual bootstrap versions of Kolmogorov-Smirnov and Cramér-von Mises type test statistics are suggested. We give proofs for bootstrap versions of the weak convergence results. The performance of the tests is demonstrated in a simulation study.

Key words and phrases: Bootstrap, empirical independence process, Kolmogorov-Smirnov test, model test, monotone rearrangements, nonparametric quantile regression, residual processes, sequential empirical process.