大协方差矩阵谱边缘的偏差校正乘子自助推断
Bias-Corrected Multiplier Bootstrap Inference for Spectral Edges of Large Covariance Matrices
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中文总结 AI 辅助
针对大协方差矩阵谱边缘推断难题，提出偏差校正乘子自助程序，通过校准乘子扰动构建置信区间并校正偏移，证明其渐近有效性，能产生无阈值的尖峰数量估计器及数据驱动的碎石图截止值。
中文摘要 AI 辅助
大协方差矩阵谱边缘的推断是高维统计中的一个基本问题。主要困难在于作为边缘自然估计量的最大非尖峰样本特征值在Tracy-Widom尺度上波动。本文提出一种偏差校正乘子自助程序用于推断体谱的确定性边缘。关键思想是引入经仔细校准的乘子扰动，将边缘波动正则化到高斯近似易于处理的稍大尺度。通过自助特征值直接构建置信区间，并进行数据驱动的重新中心化步骤以校正自助法引起的确定性边缘偏移。理论上，偏差校正和重新缩放后，最大的几个非尖峰自助特征值在数据条件下渐近高斯。基于此建立了所提置信区间的渐近有效性，其长度仅略大于Tracy-Widom尺度，并证明在替代假设下覆盖率消失，该置信区间可产生无阈值的尖峰数量估计器，无需尖峰不同或非常大，即该程序为碎石图产生数据驱动且理论合理的截止值。
英文摘要
Inference for spectral edges of large covariance matrices is a fundamental problem in high-dimensional statistics. A major difficulty is that the largest non-spiked sample eigenvalues, which serve as natural estimators of the edge, fluctuate on the Tracy--Widom scale. Consequently, valid inference requires accurate centering by the deterministic spectral edge together with a precise scaling constant, both of which are often difficult to estimate in practice under general unknown population covariance structures. In this paper, we propose a bias-corrected multiplier bootstrap procedure for inference on the deterministic edge of the bulk spectrum. The key idea is to introduce a carefully calibrated multiplier perturbation that regularizes the edge fluctuation to a slightly larger scale at which Gaussian approximation becomes tractable. The resulting confidence interval is constructed directly from bootstrap eigenvalues, together with a data-driven recentering step that corrects the bootstrap-induced shift of the deterministic edge. On the theoretical side, we show that, after bias correction and rescaling, the largest few non-spiked bootstrap eigenvalues are asymptotically Gaussian conditionally on the data. Building on this result, we establish the asymptotic validity of the proposed confidence interval, whose length is only slightly larger than the Tracy--Widom scale, and prove vanishing coverage under alternatives in which additional spikes separate from the bulk at a local scale larger than $n^{-1/6}$. As a consequence, the same confidence interval yields a threshold-free estimator for the number of spikes, without requiring the spikes to be distinct or very large. Equivalently, the procedure yields a data-driven and theoretically justified cutoff for the scree plot.
