Coverage Error Optimal Confidence Intervals for Local Polynomial Regression

We characterize the minimax bound on coverage error of Wald-type confidence intervals for nonparametric local polynomial regression. 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. In addition, we show how the large-sample interval length can be further optimized through choice of the kernel function and other tuning parameters. Our main theoretical results rely on novel Edgeworth expansions that are proven to hold uniformly over relevant classes of data generating processes. These higher-order expansions allow for the uniform kernel and any derivative order, improving on previous technical results available in the literature.