Consistency and optimality of block bootstrap schemes for distribution and variance estimation of smooth functionals of dependent data have been thoroughly investigated by Hall, Horowitz & Jing (1995), among others. However, for nonsmooth functionals, such as quantiles, much less is known. Existing results, due to Sun & Lahiri (2006), regarding strong consistency for distribution and variance estimation via the moving block bootstrap (MBB) require that b →∞, where b = n=` is the number of resampled blocks to be pasted together to form the bootstrap data series, n is the available sample size, and ` is the block length. Here we show that, in fact, weak consistency holds for any b such that 1 b=O(n=`). In other words we show that a hybrid between the subsampling bootstrap (b=1) and MBB is consistent. Empirical results illustrate the performance of hybrid block bootstrap estimators for varying numbers of blocks.

中文翻译：

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用于弱相关序列样本分位数分布和方差估计的混合块自举的一致性

Hall, Horowitz & Jing (1995) 等人已经彻底研究了用于相关数据平滑函数分布和方差估计的块自举方案的一致性和最优性。然而，对于非光滑泛函，如分位数，了解的要少得多。现有结果，由于 Sun & Lahiri (2006)，关于通过移动块引导 (MBB) 的分布和方差估计的强一致性要求 b →∞，其中 b = n=` 是要粘贴在一起的重采样块的数量为了形成引导数据系列，n 是可用样本大小，` 是块长度。在这里，我们表明，事实上，弱一致性适用于任何满足 1 b=O(n=`) 的 b。换句话说，我们表明子采样引导程序 (b=1) 和 MBB 之间的混合是一致的。