Smoothed and iterated bootstrap confidence regions for parameter vectors

The construction of confidence regions for parameter vectors is a difficult problem in the nonparametric setting, particularly when the sample size is not large. We focus on bootstrap ellipsoidal confidence regions. The bootstrap has shown promise in solving this problem, but empirical evidence often indicates that the bootstrap percentile method has difficulty in maintaining the correct coverage probability, while the bootstrap percentile-t method may be unstable, often resulting in very large confidence regions. This paper considers the smoothed and iterated bootstrap methods to construct the bootstrap percentile method ellipsoidal confidence region. The smoothed bootstrap method is based on a multivariate kernel density estimator. An optimal bandwidth matrix is established for the smoothed bootstrap procedure that reduces the coverage error of the bootstrap percentile method. We also provide an analytical adjustment to the nominal level to reduce the computational cost of the iterated bootstrap method. Simulations demonstrate that the methods can be successfully applied in practice.