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This paper studies the joint limiting behavior of extreme eigenvalues andtrace of large sample covariance matrix in a generalized spiked populationmodel, where the asymptotic regime is such that the dimension and sample sizegrow proportionally. The form of the joint limiting distribution is applied toconduct Johnson-Graybill-type tests, a family of approaches testing for signalsin a statistical model. For this, higher order correction is further made,helping alleviate the impact of finite-sample bias. The proof rests ondetermining the joint asymptotic behavior of two classes of spectral processes,corresponding to the extreme and linear spectral statistics respectively.

Asymptotic joint distribution of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model