On the Consistency of the $M\llN$ Bootstrap Approximation for a Trimmed Mean

We show that the $M\ll N$ (N is the original data sample size, M denotes the size of the bootstrap resample; $M/N\to~0$, as $M\to~\infty$) bootstrap approximation of the distribution of the trimmed mean is consistent without any conditions upon the population distribution F, whereas Efron's naive (i.e., $M=N$) bootstrap, as well as the normal approximation, fails to be consistent if the population distribution F has gaps at those two quantiles where the trimming occurs.