Local Asymptotic Normality of the spectrum of high-dimensional spiked F-ratios

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.