  A generalized spiked Fisher matrix is considered in this paper. We establish
a criterion for the description of the support of the limiting spectral
distribution of high-dimensional generalized Fisher matrix and study the almost
sure limits of the sample spiked eigenvalues where the population covariance
matrices are arbitrary which successively removed an unrealistic condition
posed in the previous works, that is, the covariance matrices are assumed to be
diagonal or diagonal block-wise structure. In addition, we also give a
consistent estimator of the population spiked eigenvalues. A series of
simulations are conducted that support the theoretical results and illustrate
the accuracy of our estimators.
