高维相依数据的中心极限定理
Central limit theorems for high dimensional dependent data
AI总结：
本文针对高维时间序列数据，在三种相依性框架下推导了高维相依随机向量之和的高斯近似非渐近误差界，提出了基于核估计的数据驱动自助法，并应用于均值检验、变点检测和置信区域构建。
摘要：
受高维时间序列数据分析中统计推断问题的启发，我们首先推导了高维相依随机向量之和在超矩形、简单凸集和稀疏凸集上高斯近似的非渐近误差界。我们在三种不同的相依性框架（$α$-混合、$m$-相依和物理相依测度）下，研究了时间相依性对向高斯随机向量收敛速度的定量影响。特别地，我们在$α$-混合框架下建立了新的误差界，并在物理相依测度下推导出了比现有结果更快的收敛速度。为了在实际统计推断问题中实现所提出的结果，我们还推导了一种基于长期协方差矩阵核估计量的数据驱动参数化自助法程序。我们将统一的高斯和自助法近似结果应用于结合$\ell^2$和$\ell^\infty$型统计量的均值向量检验、变点检测，以及协方差和精度矩阵的置信区域构建，这些应用均针对时间序列数据。
英文摘要：
Motivated by statistical inference problems in high-dimensional time series data analysis, we first derive non-asymptotic error bounds for Gaussian approximations of sums of high-dimensional dependent random vectors on hyper-rectangles, simple convex sets and sparsely convex sets. We investigate the quantitative effect of temporal dependence on the rates of convergence to a Gaussian random vector over three different dependency frameworks ($α$-mixing, $m$-dependent, and physical dependence measure). In particular, we establish new error bounds under the $α$-mixing framework and derive faster rate over existing results under the physical dependence measure. To implement the proposed results in practical statistical inference problems, we also derive a data-driven parametric bootstrap procedure based on a kernel estimator for the long-run covariance matrices. We apply the unified Gaussian and bootstrap approximation results to test mean vectors with combined $\ell^2$ and $\ell^\infty$ type statistics, change point detection, and construction of confidence regions for covariance and precision matrices, all for time series data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
