The Jackknife and the Bootstrap for General Stationary Observations

研究问题: 当观测来自平稳但存在序列依赖的过程时，如何用刀切法和自助法估计统计量的标准误？摘要关注通过连续数据块处理依赖，并考察一般统计量如何借助均值近似，在规则情形下获得渐近行为分析。
方法与思路: 刀切法依次删除每个长度为 $l$ 的连续数据块，再利用统计量重复值的样本方差；自助法则有放回地随机抽取等长观测块。摘要不将数据化约为独立同分布观测，并通过均值近似分析一般统计量的渐近行为。刀切法依次删除每个长度为 $l$ 的连续数据块，再利用统计量重复值的样本方差；自助法则有放回地随机抽取等长观测块。摘要不将数据化约为独立同分布观测，并通过均值近似分析一般统计量的渐近行为。
材料陈述的贡献: 摘要说明，均值情形的块刀切法等价于零频谱密度的加权协方差估计；在适当条件下，块长趋于无穷且相对样本长度趋于零时可获一致性。文章还给出块自助法构造，并用太阳黑子和模拟数据作说明。
阅读关注点: 重点核对摘要中“适当条件”要求何种依赖强度和统计量规则性，并跟踪块长随样本增长的两个限制。还应检查删除块后的方差归一化、均值近似误差与实际块长选择，避免只凭平稳性就认定方法有效。阅读前先备知识：建议先掌握平稳性、自协方差、序列相关与长程方差，理解零频谱密度为何与样本均值的方差有关。再复习普通刀切法、自助法及统计量的渐近线性近似，便于理解连续数据块如何保留重抽样所需的依赖信息。
阅读边界: 以下限制对应原请求所用摘要版本：基于原文摘要，未阅读全文。摘要给出块构造、块长增长要求及一致性结论，但未列全依赖条件、一般统计量的规则性要求和有限样本表现；阅读建议涉及的长程方差与渐近线性近似属于背景知识。经AI辅助核对旧与当前摘要内容等价；未另读刊稿全文，不能据此证明整个稿件版本等价。

GPT-6.1 Sol · 极高 · 2026-10-01

原文摘要

We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero. Under appropriate conditions consistency is obtained if $l = l(n) \rightarrow \infty$ and $l(n)/n \rightarrow 0$ . General statistics are approximated by an arithmetic mean. In regular cases this approximation determines the asymptotic behavior. Bootstrap replicates are constructed by selecting blocks of length $l$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero. Under appropriate conditions consistency is obtained if $l = l(n) \rightarrow \infty$ and $l(n)/n \rightarrow 0$. General statistics are approximated by an arithmetic mean. In regular cases this approximation determines the asymptotic behavior. Bootstrap replicates are constructed by selecting blocks of length $l$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）

笔记与分组

分类依据与纠错

The Jackknife and the Bootstrap for General Stationary Observations

代码、数据与补充材料

一般平稳观测下的刀切法与自助法

原文摘要

同方向的基础文献

同方向的基础文献