Spiked eigenvalues of noncentral Fisher matrix with applications

Abstract

In this paper, we investigate the asymptotic behavior of spiked eigenvalues of the noncentral Fisher matrix defined by ${\mathbf F}_p={\mathbf C}_n(\mathbf S_N)^{-1}$, where ${\mathbf C}_n$ is a noncentral sample covariance matrix defined by $(\mathbf \Xi+\mathbf X)(\mathbf \Xi+\mathbf X)^*/n$ and $\mathbf S_N={\mathbf Y}{\mathbf Y}^*/N$. The matrices $\mathbf X$ and $\mathbf Y$ are two independent {Gaussian} arrays, with respective $p\times n$ and $p\times N$ and the Gaussian entries of them are \textit {independent and identically distributed} (i.i.d.) with mean $0$ and variance $1$. When $p$, $n$, and $N$ grow to infinity proportionally, we establish a phase transition of the spiked eigenvalues of $\mathbf F_p$. Furthermore, we derive the \textit{central limiting theorem} (CLT) for the spiked eigenvalues of $\mathbf F_p$. As an accessory to the proof of the above results, the fluctuations of the spiked eigenvalues of ${\mathbf C}_n$ are studied, which should have its own interests. Besides, we develop the limits and CLT for the sample canonical correlation coefficients by the results of the spiked noncentral Fisher matrix and give three consistent estimators, including the population spiked eigenvalues and the population canonical correlation coefficients.