We consider the problem of testing monotonicity of the regression function in a nonparametric regression model. We introduce test statistics that are functionals of a certain natural $U$-process. We study the limiting distribution of these test statistics through strong approximation methods and the extreme value theory for Gaussian processes. We show that the tests are consistent against general alternatives.
"Testing monotonicity of regression." Ann. Statist. 28 (4) 1054 - 1082, August 2000. https://doi.org/10.1214/aos/1015956707