A CLT for the LSS of large-dimensional sample covariance matrices with diverging spikes
含发散尖峰的大维样本协方差矩阵线性谱统计量的中心极限定理AI 解读 ↓
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含发散尖峰的大维样本协方差矩阵线性谱统计量的中心极限定理
- 研究问题
- 当总体协方差矩阵的尖峰特征值随维度发散时，线性谱统计量如何获得极限分布，并用于协方差结构检验？
- 方法与思路
- 作者扩展线性谱统计量的中心极限定理，容许有界或发散尖峰，并推导修正似然比与迹检验在备择假设下的渐近分布。
- 材料陈述的贡献
- 摘要称，新极限方差同时涉及尖峰与主体特征值；除单尖峰情形外，所考察谱统计量可能比最大根检验具有更高渐近功效。
- 阅读关注点
- 应核对维度与样本量的增长关系、矩条件、尖峰发散速度及统计量修正方式，并确认功效比较适用的备择范围。
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- 仅据所给 OpenAlex 索引摘要，稿件版本未注明；具体正则条件与功效比较范围未提供，不能推定刊本变化或普遍优越性。
原文摘要
In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSSs) of a large-dimensional sample covariance matrix when the population covariance matrices are involved with diverging spikes. This constitutes a nontrivial extension of the Bai–Silverstein theorem (BST) (Ann. Probab. 32 (2004) 553–605), a theorem that has strongly influenced the development of high-dimensional statistics, especially in the applications of random matrix theory to statistics. Recently, there has been a growing realization that the assumption of uniform boundedness of the population covariance matrices in the BST is not satisfied in some fields, such as economics, where the variances of principal components may diverge as the dimension tends to infinity. Therefore, in this paper, we aim to eliminate this obstacle to applications of the BST. Our new CLT accommodates spiked eigenvalues, which may either be bounded or tend to infinity. A distinguishing feature of our result is that the variance in the new CLT is related to both spiked eigenvalues and bulk eigenvalues, with dominance being determined by the divergence rate of the largest spiked eigenvalues. The new CLT for LSS is then applied to test the hypothesis that the population covariance matrix is the identity matrix or a generalized spiked model. The asymptotic distributions of the corrected likelihood ratio test statistic and the corrected Nagao’s trace test statistic are derived under the alternative hypothesis. Moreover, we present power comparisons between these two LSSs and Roy’s largest root test. In particular, we demonstrate that except for the case in which there is only one spike, the LSSs could exhibit higher asymptotic power than Roy’s largest root test.