Abstract: In this work, we study the diagnostics of parametric regression models when both the response variable and the covariates are distorted by errors. We employ a projected empirical process to develop Cramér-von Mises and Kolmogorov-Smirnov tests with dimension-reduction effects. We apply random approximation to enable an expedient calculation of the Kolmogorov-Smirnov test for checking the suitability of regression models. The proposed tests are shown to be consistent and can detect an alternative hypothesis close to the null hypothesis at the root-n rate. Simulation studies show that the proposed tests outperform existing methods. A real data set is analyzed for illustration.

Key words and phrases: Cramér-von Mises test, dimension-reduction, empirical process, Kolmogorov-Smirnov test, random approximation.