The resampling method via representative points

The bootstrap method relies on resampling from the empirical distribution to provide inferences about the population with a distribution F. The empirical distribution serves as an approximation to the population. It is possible, however, to resample from another approximating distribution of F to conduct simulation-based inferences. In this paper, we utilize representative points to form an alternative approximating distribution of F for resampling. The representative points in terms of minimum mean squared error from F have been widely applied to numerical integration, simulation, and the problems of grouping, quantization, and classification. The method of resampling via representative points can be used to estimate the sampling distribution of a statistic of interest. A basic theory for the proposed method is established. We prove the convergence of higher-order moments of the new approximating distribution of F, and establish the consistency of sampling distribution approximation in the cases of the sample mean and sample variance under the Kolmogorov metric and Mallows–Wasserstein metric. Based on some numerical studies, it has been shown that the proposed resampling method improves the nonparametric bootstrap in terms of confidence intervals for mean and variance.