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In this paper, we study limiting laws and consistent estimation criteria forthe extreme eigenvalues in a spiked covariance model of dimension $p$. Firstly,for fixed $p$, we propose a generalized estimation criterion that canconsistently estimate, $k$, the number of spiked eigenvalues. Compared with theexisting literature, we show that consistency can be achieved under weakerconditions on the penalty term. Next, allowing both $p$ and $k$ to diverge, wederive limiting distributions of the spiked sample eigenvalues using randommatrix theory techniques. Notably, our results do not require the spikedeigenvalues to be uniformly bounded from above or tending to infinity, as havebeen assumed in the existing literature. Based on the above derived results, weformulate a generalized estimation criterion and show that it can consistentlyestimate $k$, while $k$ can be fixed or grow at an order of $k=o(n^{1/3})$. Wefurther show that the results in our work continue to hold under a generalpopulation distribution without assuming normality. The efficacy of theproposed estimation criteria is illustrated through comparative simulationstudies.

Limiting laws and consistent estimation criteria for fixed and diverging number of spiked eigenvalues