Detection of the number of principal components by extended AIC-type method

Estimating the number of principal components is one of the fundamental problems in many scientific fields such as signal processing (or the spiked covariance model). In this paper, we first demonstrate that, for fixed $p$, any penalty term of the form $k'(p-k'/2+1/2)C_n$ may lead to an asymptotically consistent estimator under the condition that $C_n\to\infty$ and $C_n/n\to0$. We also extend our results to the case $n,p\to\infty$, with $p/n\to c>0$. In this case, for $k=o(n^{\frac{1}{3}})$, we first investigate the limiting laws for the leading eigenvalues of the sample covariance matrix $S_n$ under the condition that $\lambda_k>1+\sqrt{c}$. At low SNR, since the AIC tends to underestimate the number of signals $k$, the AIC should be re-defined in this case. As a natural extension of the AIC for fixed $p$, we propose the extended AIC (EAIC), i.e., the AIC-type method with tuning parameter $\gamma=\varphi(c)=1/2+\sqrt{1/c}-\log(1+\sqrt{c})/c$, and demonstrate that the EAIC-type method, i.e., the AIC-type method with tuning parameter $\gamma>\varphi(c)$, can select the number of signals $k$ consistently. In the following two cases, (1) $p$ fixed, $n\to\infty$, (2) $n,p\to\infty$ with $p/n\to 0$, if the AIC is defined as the degeneration of the EAIC in the case $n,p\to\infty$ with $p/n\to c>0$, i.e., $\gamma=\lim_{c\rightarrow 0+0}\varphi(c)=1$, then we have essentially demonstrated that, to achieve the consistency of the AIC-type method in the above two cases, $\gamma>1$ is required. Moreover, we show that the EAIC-type method is essentially tuning-free and outperforms the well-known KN estimator proposed in Kritchman and Nadler (2008) and the BCF estimator proposed in Bai, Choi and Fujikoshi (2018). Numerical studies indicate that the proposed method works well.