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   .
This paper studies higher-order inference properties of nonparametric local polynomial regression methods under random sampling. We prove Edgeworth expansions for t statistics and coverage error expansions for interval estimators that (i) hold uniformly in the data generating process, (ii) allow for the uniform kernel, and (iii) cover estimation of derivatives of the regression function. The terms of the higher-order expansions, and their associated rates as a function of the sample size and bandwidth sequence, depend on the smoothness of the population regression function, the smoothness exploited by the inference procedure, and on whether the evaluation point is in the interior or on the boundary of the support . We prove that robust bias corrected confidence intervals have the fastest coverage error decay rates in all cases, and we use our results to deliver novel, inference-optimal bandwidth selectors. The main methodological results are implemented in companionR andStata software packages .
R
We characterize the minimax bound on coverage error of Wald-type confidence intervals for
This paper studies higher-order inference properties of
R
. This bound depends
methods under random sampling. We prove Edgeworth expansions for t statistics and coverage error expansions for interval estimators that (i) hold uniformly in the data generating process, (ii) allow for the uniform kernel, and (iii) cover estimation of derivatives of the regression function. The terms of the higher-order expansions, and their associated rates as a function of the sample size and bandwidth sequence, depend
D
of interest
R
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
. We prove
R
are able to attain the minimax rate when coupled with the principled,
have the fastest coverage error decay rates in all cases, and we use our results to deliver novel,
R
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
bandwidth selectors. The main methodological results are implemented in companion
A
R
A
and
A
Stata
A
software packages