Block-based resampling estimators have been intensively investigated for weakly dependent time processes, which has helped to inform implementation (e.g., best block sizes). However, little is known about resampling performance and block sizes under strong or long-range dependence. To establish guideposts in block selection, we consider a broad class of strongly dependent time processes, formed by a transformation of a stationary long-memory Gaussian series, and examine block-based resampling estimators for the variance of the prototypical sample mean; extensions to general statistical functionals are also considered. Unlike weak dependence, the properties of resampling estimators under strong dependence are shown to depend intricately on the nature of non-linearity in the time series (beyond Hermite ranks) in addition the long-memory coefficient and block size. Additionally, the intuition has often been that optimal block sizes should be larger under strong dependence (say $O(n^{1/2})$ for a sample size $n$) than the optimal order $O(n^{1/3})$ known under weak dependence. This intuition turns out to be largely incorrect, though a block order $O(n^{1/2})$ may be reasonable (and even optimal) in many cases, owing to non-linearity in a long-memory time series. While optimal block sizes are more complex under long-range dependence compared to short-range, we provide a consistent data-driven rule for block selection, and numerical studies illustrate that the guides for block selection perform well in other block-based problems with long-memory time series, such as distribution estimation and strategies for testing Hermite rank.

中文翻译：

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高斯从属长程依赖过程的最优块重采样

基于块的重采样估计器已经针对弱依赖时间过程进行了深入研究，这有助于为实现提供信息（例如，最佳块大小）。然而，对于强或长期依赖下的重采样性能和块大小知之甚少。为了在块选择中建立指南，我们考虑了由固定长记忆高斯级数变换形成的一类广泛的强相关时间过程，并检查基于块的重采样估计器的原型样本均值的方差；还考虑了对一般统计泛函的扩展。与弱依赖不同，除了长记忆系数和块大小之外，强依赖下的重采样估计器的属性被证明与时间序列中的非线性性质（超出 Hermite 等级）有着错综复杂的关系。此外，直觉通常认为，在强依赖性（例如样本大小为 $n$ 的 $O(n^{1/2})$）下，最佳块大小应该大于最佳顺序 $O(n^{1 /3})$ 在弱依赖下已知。这种直觉在很大程度上是不正确的，尽管由于长记忆时间序列中的非线性，在许多情况下块顺序 $O(n^{1/2})$ 可能是合理的（甚至是最优的）。虽然与短程相比，长程依赖下的最佳区块大小更复杂，但我们为区块选择提供了一致的数据驱动规则，