  We consider two types of spiked multivariate F distributions: a scaled
distribution with the scale matrix equal to a rank-one perturbation of the
identity, and a distribution with trivial scale, but rank-one non-centrality.
The norm of the rank-one matrix (spike) parameterizes the joint distribution of
the eigenvalues of the corresponding F matrix. We show that, for a spike
located above a phase transition threshold, the asymptotic behavior of the log
ratio of the joint density of the eigenvalues of the F matrix to their joint
density under a local deviation from this value depends only on the largest
eigenvalue $\lambda_{1}$. Furthermore, $\lambda_{1}$ is asymptotically normal,
and the statistical experiment of observing all the eigenvalues of the F matrix
converges in the Le Cam sense to a Gaussian shift experiment that depends on
the asymptotic mean and variance of $\lambda_{1}$. In particular, the best
statistical inference about a sufficiently large spike in the local asymptotic
regime is based on the largest eigenvalue only. As a by-product of our
analysis, we establish joint asymptotic normality of a few of the largest
eigenvalues of the multi-spiked F matrix when the corresponding spikes are
above the phase transition threshold.
