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-k perturbation of the identity, and a distribution with trivial scale, but rank-k non-centrality. The eigenvalues of the rank-k matrix (spikes) parameterize the joint distribution of the eigenvalues of the corresponding F matrix. We show that, for the spikes 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 these values depends only on the k of the largest eigenvalues {code} Furthermore, we show that {code} are asymptotically jointly 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 means and variances of {code}. In particular, the best statistical inference about sufficiently large spikes in the local asymptotic regime is based on the k of the largest eigenvalues only.