We consider the problem of inference on a regression function at a point when the entire function satisfies a sign or shape restriction under the null. We propose a test that achieves the optimal minimax rate adaptively over a range of Hölder classes, up to a $\log\log n$ term, which we show to be necessary for adaptation. We apply the results to adaptive one-sided tests for the regression discontinuity parameter under a monotonicity restriction, the value of a monotone regression function at the boundary and the proportion of true null hypotheses in a multiple testing problem.
"Adaptive testing on a regression function at a point." Ann. Statist. 43 (5) 2086 - 2101, October 2015. https://doi.org/10.1214/15-AOS1342