On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Estimation∗

This paper studies the effect of bias correction on confidence interval estimators in the context of kernel-based nonparametric density estimation. We consider explicit plug-in bias correction but, in contrast to standard approaches, we allow the bias estimator to (potentially) have a first-order impact on the distributional approximation. This approach is meant to more accurately capture the finite-sample behavior of the estimator and corresponding confidence interval. We propose a new standard error based on this asymptotic experiment, which allows greater flexibility in bias estimation. Bias reduction can alternatively be achieved implicitly by undersmoothing the density estimator, and we seek to compare these two approaches, as investigated by Hall (1992. Effect of Bias Estimation on Coverage Accuracy of Bootstrap Confidence Intervals for a Probability Density. Annals of Statistics 20, 675–694). Using Edgeworth expansions, we show that explicit bias correction, coupled with our proposed standard errors, may outperform undersmoothing in terms of confidence interval coverage error. This implies that undersmoothing does not strictly dominate bias correction, as was previously found by Hall (1992). We use our results to discuss interval length implications and to give new bandwidth selection guides for density estimation and bias correction. We also outline briefly how our findings extend to other nonparametric contexts. ∗We thank Michael Jansson for comments and suggestions.