Effect of Bias Estimation on Coverage Accuracy of Bootstrap Confidence Intervals for a Probability Density

The bootstrap is a poor estimator of bias in problems of curve estimation, and so bias must be corrected by other means when the bootstrap is used to construct confidence intervals for a probability density. Bias may either be estimated explicitly, or allowed for by undersmoothing the curve estimator. Which of these two approaches is to be preferred? In the present paper we address this question from the viewpoint of coverage accuracy, assuming a given number of derivatives of the unknown density. We conclude that the simpler, undersmoothing method is more efficacious. Undersmoothing also has advantages from the standpoint of minimizing interval width. We derive formulae for bandwidths which are optimal in terms of coverage accuracy and also give formulae for the coverage error which results from using those bandwidths.