This paper investigates the effects of smoothed bootstrap iterations on coverage probabilities of smoothed bootstrap and bootstrap-t confidence intervals for population quantiles, and establishes the optimal kernel bandwidths at various stages of the smoothing procedures. The conventional smoothed bootstrap and bootstrap-t methods have been known to yield one-sided coverage errors of orders O(n−1/2) and o(n−2/3), respectively, for intervals based on the sample quantile of a random sample of size n. We sharpen the latter result to O(n−5/6) with proper choices of bandwidths at the bootstrapping and Studentization steps. We show further that calibration of the nominal coverage level by means of the iterated bootstrap succeeds in reducing the coverage error of the smoothed bootstrap percentile interval to the order O(n−2/3) and that of the smoothed bootstrap-t interval to O(n−58/57), provided that bandwidths are selected of appropriate orders. Simulation results confirm our asymptotic findings, suggesting that the iterated smoothed bootstrap-t method yields the most accurate coverage. On the other hand, the iterated smoothed bootstrap percentile method interval has the advantage of being shorter and more stable than the bootstrap-t intervals.
"Iterated smoothed bootstrap confidence intervals for population quantiles." Ann. Statist. 33 (1) 437 - 462, February 2005. https://doi.org/10.1214/009053604000000878