Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices
By Tommaso Cai, Xiao Han, Guangming Pan. Annals of Statistics, 2017
87 citations recorded by the source corpus.
We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance matrix model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy-Widom law for the largest nonspiked sample eigenvalue is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a Central Limit Theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.
Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/81e25813b9e75453d1620c0b0a4425a6f849cfc9
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In Artificial Intelligence
In Mathematical Physics
In Statistics and Probability
- Jianqing Fan
- Shu-rong Zheng
- Jianhua Guo
- Cun-Hui Zhang
- Jinchi Lv
- Hansheng Wang
- Tommaso Cai
- Yingcun Xia
Citations in the corpus
- Estimating Number of Factors by Adjusted Eigenvalues Thresholding (2019, methodology): “Based on a confi-dence interval of the largest non-spiked eigenvalue of the estimated covariance matrix, Cai et al. (2017) proposed an algorithm to determine the number of common factors under the convergence regime that the dimension and the sample size tend to infinity proportionally.”