Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications
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高维 U 统计量的高斯与自助法近似及其应用
- 研究问题
- 当维度远大于样本量时，如何近似非退化 U 统计量落入高维超矩形的概率，并为最大值和矩阵元素构造推断？
- 方法与思路
- 作者提出两步高斯近似，并统一分析经验自助法、随机重加权自助法及使用删一法协方差估计的高斯乘子自助法。
- 材料陈述的贡献
- 摘要称三类自助法具有相同的一致收敛速率，并给出最大值分位数近似、协方差阈值选择及矩阵同时推断应用。
- 阅读关注点
- 核对非退化条件、核函数矩条件与维度增长限制，以及三类方法的一阶等价和阈值误差改进分别适用哪些数据类别。
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- 仅据 OpenAlex 索引摘要，稿件版本未注明；具体条件、常数与证明未提供，不能把特定数据类别中的误差改进推广为普遍优势。
原文摘要
This paper studies the Gaussian and bootstrap approximations for the probabilities of a nondegenerate U-statistic belonging to the hyperrectangles in $\mathbb{R}^{d}$ when the dimension $d$ is large. A two-step Gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed. Subject to mild moment conditions on the kernel, we establish the explicit rate of convergence uniformly in the class of all hyperrectangles in $\mathbb{R}^{d}$ that decays polynomially in sample size for a high-dimensional scaling limit, where the dimension can be much larger than the sample size. We also provide computable approximation methods for the quantiles of the maxima of centered U-statistics. Specifically, we provide a unified perspective for the empirical bootstrap, the randomly reweighted bootstrap and the Gaussian multiplier bootstrap with the jackknife estimator of covariance matrix as randomly reweighted quadratic forms and we establish their validity. We show that all three methods are inferentially first-order equivalent for high-dimensional U-statistics in the sense that they achieve the same uniform rate of convergence over all $d$-dimensional hyperrectangles. In particular, they are asymptotically valid when the dimension $d$ can be as large as $O(e^{n^{c}})$ for some constant $c\in(0,1/7)$. The bootstrap methods are applied to statistical applications for high-dimensional non-Gaussian data including: (i) principled and data-dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals; (ii) simultaneous inference for the covariance and rank correlation matrices. In particular, for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter, we show that for a class of sub-Gaussian data, error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold. In addition, we also show that the Gaussian-like convergence rates can be achieved for heavy-tailed data, which are less conservative than those obtained by the Bonferroni technique that ignores the dependency in the underlying data distribution.