  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.
