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This paper novelly transforms lack-of-fit tests for parametric quantile regression models into checking the equality of two conditional distributions of covariates. We then can borrow these successful test statistics from the rich literature of two-sample problems, and this gives us much flexibility in constructing a reliable test according our experiences on covariates.
As an illustration, three two-sample test statistics are considered, and it will lead to He and Zhu's (2003) test when a Cramer-von Mises test statistic is employed.
The second one is a practical two-sample test statistic especially for multivariate distributions, and the resulting test is still applicable in real applications when the number of covariates is moderate or even large. In the last case, we provide a lack-of-fit test based on two-sample tests on moments for the high dimensional data.
Its usefulness is demonstrated by simulation experiments and a real example. | 
