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. The bootstrap has shown promise in solving this problem, but empirical evidence often indicates that some bootstrap methods have difficulty in maintaining the correct coverage probability, while other methods may be unstable, often resulting in very large confidence regions. One way to improve the performance of a bootstrap confidence region is to restrict the shape of the region in such a way that the error term of an expansion is as small an order as possible. To some extent, this can be achieved by using the bootstrap to construct an ellipsoidal confidence region. This paper studies the effect of using the smoothed and iterated bootstrap methods to construct an ellipsoidal confidence region for a parameter vector. The smoothed estimate is based on a multivariate kernel density estimator. This paper establishes a bandwidth matrix for the smoothed bootstrap procedure that reduces the asymptotic coverage error of the bootstrap percentile method ellipsoidal confidence region. 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.