Abstract
Under fairly general assumptions on the underlying distribution function, the bootstrap process, pertaining to the sample $q$-quantile, converges weakly in $D_\mathbb{R}$ to the standard Brownian motion. Furthermore, weak convergence of a smoothed bootstrap quantile estimate is proved which entails that in this particular case the smoothed bootstrap estimate outperforms the nonsmoothed one.
Citation
M. Falk. R.-D. Reiss. "Weak Convergence of Smoothed and Nonsmoothed Bootstrap Quantile Estimates." Ann. Probab. 17 (1) 362 - 371, January, 1989. https://doi.org/10.1214/aop/1176991515
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