View Submission - CFE

A0990
**Title: **Local asymptotic normality of the spectrum of high-dimensional spiked $F$-ratios
**Authors: **Alexei Onatski - University of Cambridge (United Kingdom) **[presenting]**

**Abstract: **Two types of spiked multivariate $F$ distributions are considered: 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 $\lambda_{1},...,\lambda_{k}$. Furthermore, we show that $\lambda_{1},...,\lambda_{k}$ 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 $\lambda_{1},...,\lambda_{k}$. 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.