Consider two $p$-variate populations, not necessarily Gaussian, with covariance matrices $Σ_1$ and $Σ_2$, respectively, and let $S_1$ and $S_2$ be the sample covariances matrices from samples of the populations with degrees of freedom $T$ and $n$, respectively. When the difference $Δ$ between $Σ_1$ and $Σ_2$ is of small rank compared to $p,T$ and $n$, the Fisher matrix $F=S_2^{-1}S_1$ is called a {\em spiked Fisher matrix}. When $p,T$ and $n$ grow to infinity proportionally, we establish a phase transition for the extreme eigenvalues of $F$: when the eigenvalues of $Δ$ ({\em spikes}) are above (or under) a critical value, the associated extreme eigenvalues of the Fisher matrix will converge to some point outside the support of the global limit (LSD) of other eigenvalues; otherwise, they will converge to the edge points of the LSD. Furthermore, we derive central limit theorems for these extreme eigenvalues of the spiked Fisher matrix. The limiting distributions are found to be Gaussian if and only if the corresponding population spike eigenvalues in $Δ$ are {\em simple}. Numerical examples are provided to demonstrate the finite sample performance of the results. In addition to classical applications of a Fisher matrix in high-dimensional data analysis, we propose a new method for the detection of signals allowing an arbitrary covariance structure of the noise. Simulation experiments are conducted to illustrate the performance of this detector.
Spectrally-Corrected and Regularized QDA Classifier for Spiked Covariance Model
School of Data Sciences, Zhejiang University of Finance and Economics（浙江财经大学数据科学学院）；School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）；Northeast Normal University（东北师范大学）
Limiting laws and consistent estimation criteria for fixed and diverging number of spiked eigenvalues
School of Mathematics and Statistics, Beijing Technology and Business University（北京工商大学数学与统计学院）；Department of Statistics, University of Michigan（密歇根大学统计学系）；Department of Statistics and Key Lab NAA–MOE, Central China Normal University（华中师范大学统计学系与国家数理分析重点实验室）；Goizueta Business School, Emory University（埃默里大学戈伊泽塔商学院）
