Bootstrap Choice of Bandwidth for Density Estimation

Abstract

A bootstrap-based choice of bandwidth for kernel density estimation is introduced. The method works by estimating the integrated mean squared error (IMSE) for any given bandwidth and then minimizing over all bandwidths. A straightforward application of the bootstrap method to estimate the IMSE fails because it does not capture the bias component. A smoothed bootstrap method based on an initial density estimate is described that solves this problem. It is possible to construct pointwise and simultaneous confidence intervals for the density. The simulation study compares cross-validation and the bootstrap method over a wide range of densities—a long-tailed, a short-tailed, an asymmetric, and a bimodal, among others. The bootstrap method uniformly outperforms cross-validation. The accuracy of the constructed confidence bands improves as the sample size increases.

Key Words:

• Cross-validation
• Confidence bands
• Kernel density estimation
• Smoothed bootstrap