极值特征值的极限定律与一致估计准则：固定和发散的突出特征值数量
Limiting laws and consistent estimation criteria for fixed and diverging number of spiked eigenvalues
AI总结：
本文研究了高维数据中极值特征值的极限定律与一致估计准则。针对固定维度p，提出了一种通用估计准则以一致估计突出特征值数量k。在允许p和k发散的情况下，利用随机矩阵理论推导了突出样本特征值的极限分布，无需假设突出特征值有上界或趋于无穷。通过模拟研究验证了所提准则的有效性。
摘要：
在本文中，我们研究了高维数据中极值特征值的极限定律与一致估计准则。首先，对于固定p，我们提出了一种通用估计准则，可以一致估计k，即突出特征值的数量。与现有文献相比，我们证明在更弱的惩罚项条件下可以实现一致性。接下来，允许p和k发散，我们利用随机矩阵理论技术推导了突出样本特征值的极限分布。值得注意的是，我们的结果不要求突出特征值有上界或趋于无穷，这与现有文献中的假设不同。基于上述推导结果，我们提出了一种通用估计准则，并证明其可以一致估计k，而k可以固定或以k=o(n^{1/3})的速度增长。我们进一步证明，在一般总体分布下，本文结果仍然成立，而无需假设正态性。所提估计准则的有效性通过比较模拟研究加以验证。
英文摘要：
In this paper, we study limiting laws and consistent estimation criteria for the extreme eigenvalues in a spiked covariance model of dimension $p$. Firstly, for fixed $p$, we propose a generalized estimation criterion that can consistently estimate, $k$, the number of spiked eigenvalues. Compared with the existing literature, we show that consistency can be achieved under weaker conditions on the penalty term. Next, allowing both $p$ and $k$ to diverge, we derive limiting distributions of the spiked sample eigenvalues using random matrix theory techniques. Notably, our results do not require the spiked eigenvalues to be uniformly bounded from above or tending to infinity, as have been assumed in the existing literature. Based on the above derived results, we formulate a generalized estimation criterion and show that it can consistently estimate $k$, while $k$ can be fixed or grow at an order of $k=o(n^{1/3})$. We further show that the results in our work continue to hold under a general population distribution without assuming normality. The efficacy of the proposed estimation criteria is illustrated through comparative simulation studies.
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