Coverage Accuracy in Bias-Corrected Nonparametric and Linear Semiparametric Inference

Abstract

This talk addresses tuning parameter choice and higher-order refinements in nonparametric and semiparametric inference. In the first part, we study the effects of bias correction on confidence interval coverage for nonparametric kernel density and local polynomial regression estimation. We show formally that bias correction may be preferred to undersmoothing when the goal is to minimize the coverage error of the confidence interval. This result is established using a novel, yet simple, studentization approach for inference, which leads to a new way of constructing kernel-based nonparametric confidence intervals. Bandwidth selection is discussed, and shown to be very simple to implement. Indeed, we show that MSE-optimal bandwidths deliver the fastest coverage error rates when second-order kernels are employed. Next, we investigate a class of "linear" semiparametric kernel-based studentized statistics and demonstrate the higher-order refinements offered by the so-called "small bandwidth" asymptotic approach relative to conventional asymptotics. In particular, for semiparametric density-weighted average derivatives we show that the rate of approximation no longer depends on the dimension of the data nor the bandwidth, and moreover, that the parametric rate automatically (i.e. for any bandwidth), provided mild bias conditions are satisfied. The efficiency loss of the small bandwidth approach, measured by interval length, is quite small. A new bandwidth selector is discussed.