2014/11/14 by Prathapasinghe Dharmawansa, Iain M. Johnstone, Dharmawansa, Prathapasinghe +3
Mathematics · #62B15 #62F12 #62H12 #62H15 #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #msc:62B15 #msc:62F12 #msc:62H12 #msc:62H15 #stat.TH
paper · pdf · doi:10.48550/arxiv.1411.3875
42 pages, 2 figures
arxiv created 2014/11/14 · arxiv updated 2014/11/17
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 λ1. Furthermore, λ1 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 λ1. 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.