On Convergence of Different Bootstrap Approximations for Medians

This study aims to investigate the convergence properties of the exact closed-form formula for the distribution of indices in a sorted sample being medians of Poisson-bootstrapped subsamples toward the binomial distribution. We find that although the two distributions do not coincide exactly, their asymptotic behaviors are of the same order, specifically O(1/√n) as the sample size n increases. The findings have significant implications for statistical inference and resampling techniques, as they provide deeper insights into the efficiency and accuracy of the Poisson bootstrap method.