We derive Bahadur-type representations for quantile estimates obtained from two different types of nonparametric bootstrap resampling--the commonly used uniform resampling method, where each sample value is drawn with the same probability, and importance resampling, where different sample values are assigned different resampling weights. These results are applied to obtain the relative efficiency of uniform resampling and importance resampling and to derive exact convergence rates, both weakly and strongly, for either type of resampling.
"Bahadur Representations for Uniform Resampling and Importance Resampling, with Applications to Asymptotic Relative Efficiency." Ann. Statist. 19 (2) 1062 - 1072, June, 1991. https://doi.org/10.1214/aos/1176348138