  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.
