Before revision:  This bound depends on the smoothness of the population regression function, the smoothness exploited by the inference procedure, and on whether the evaluation point of interest is in the interior or on the boundary of the support of the regression function. Our results also cover inference on derivatives of the regression function, in which case we find that the minimax coverage error bound does not depend on the order of the derivative being estimated. We show that robust bias corrected confidence intervals are able to attain the minimax rate when coupled with the principled, inference-optimal tuning parameter selections we propose.
After revision:  This bound depends on the smoothness of the population regression function, the smoothness exploited by the inference procedure, and on whether the evaluation point of interest is in the interior or on the boundary of the support . We prove that robust bias corrected confidence intervals are able to attain the minimax rate when coupled with the principled, inference-optimal tuning parameter selections we propose.