This paper studies the joint limiting behavior of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model, where the asymptotic regime is such that the dimension and sample size grow proportionally. The form of the joint limiting distribution is applied to conduct Johnson–Graybill-type tests, a family of approaches testing for signals in a statistical model. For this, higher order correction is further made, helping alleviate the impact of finite-sample bias. The proof rests on determining the joint asymptotic behavior of two classes of spectral processes, corresponding to the extreme and linear spectral statistics, respectively.
Zeng Li. Fang Han. Jianfeng Yao. "Asymptotic joint distribution of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model." Ann. Statist. 48 (6) 3138 - 3160, December 2020. https://doi.org/10.1214/19-AOS1882