Summary

We establish a general theory of optimality for block bootstrap distribution estimation for sample quantiles under mild strong mixing conditions. In contrast to existing results, we study the block bootstrap for varying numbers of blocks. This corresponds to a hybrid between the sub- sampling bootstrap and the moving block bootstrap, in which the number of blocks is between 1 and the ratio of sample size to block length. The hybrid block bootstrap is shown to give theoretical benefits, and startling improvements in accuracy in distribution estimation in important practical settings. The conclusion that bootstrap samples should be of smaller size than the original sample has significant implications for computational efficiency and scalability of bootstrap methodologies with dependent data. Our main theorem determines the optimal number of blocks and block length to achieve the best possible convergence rate for the block bootstrap distribution estimator for sample quantiles. We propose an intuitive method for empirical selection of the optimal number and length of blocks, and demonstrate its value in a nontrivial example.

中文翻译：

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具有相关数据的样本分位数的块引导最优性和经验块选择

概括

我们为温和强混合条件下的样本分位数建立了块自举分布估计的一般最优理论。与现有结果相反，我们研究了不同块数的块引导程序。这对应于子采样自举和移动块自举之间的混合，其中块的数量在 1 和样本大小与块长度的比率之间。混合块引导程序显示出理论优势，并在重要的实际设置中对分布估计的准确性进行了惊人的改进。bootstrap 样本的大小应该小于原始样本的结论对具有相关数据的 bootstrap 方法的计算效率和可扩展性具有重要意义。我们的主要定理确定了最佳块数和块长度，以实现样本分位数的块自举分布估计器的最佳收敛速度。我们提出了一种经验选择最佳块数和长度的直观方法，并在一个非平凡的例子中证明了它的价值。