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.

翻译：本文研究维数为$p$的尖峰协方差模型中极端特征值的极限定律与一致估计准则。首先，对于固定$p$，我们提出了一种广义估计准则，能够一致估计尖峰特征值的个数$k$。与现有文献相比，我们证明在更弱的惩罚项条件下可实现一致性。其次，允许$p$和$k$同时发散，我们利用随机矩阵理论技术推导了尖峰样本特征值的极限分布。值得注意的是，我们的结果不要求尖峰特征值必须一致上有界或趋向无穷——而现有文献中常作此假设。基于上述推导结果，我们构建了广义估计准则，并证明其能一致估计$k$，其中$k$可固定或按$k=o(n^{1/3})$阶增长。进一步表明，我们的结果在无需正态性假设的一般总体分布下仍然成立。通过比较模拟研究验证了所提估计准则的有效性。