Asymptotic normality and analysis of variance of log-likelihood ratios in spiked random matrix models

TL;DR

This paper analyzes the asymptotic behavior of log-likelihood ratios in spiked random matrix models, providing normality results and variance decompositions to improve understanding of signal detection in high-dimensional settings.

Contribution

It extends analysis to multi-spiked models with dependent priors and derives asymptotic normality and variance decompositions for the log-likelihood ratios.

Findings

Log-likelihood ratios are asymptotically normal below certain SNR thresholds.

Variances of log-likelihood ratios decompose into sums of bipartite signed cycle statistics.

Results apply to Gaussian mixtures and spiked Wishart models.

Abstract

The present manuscript studies signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on the signal component that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-to- noise ratios are below certain thresholds. In addition, we show that the variances of the log-likelihood ratios can be asymptotically decomposed as the sums of those of a collection of statistics which we call bipartite signed cycles.