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-r 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 λ1, ..., λk. Furthermore, we show that λ1, ..., λk 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 λ1, ..., λk. In particular, the best statistical inference about suffi ciently large spikes in the local asymptotic regime is based on the k of the largest eigenvalues only.