🤖 AI Summary
This paper addresses the reduced statistical power of Hotelling’s $T^2$ test in high-dimensional, low-sample-size settings ($p/n o gamma > 0$), where inaccurate covariance matrix estimation severely impairs performance. We propose a nonlinear covariance shrinkage estimator that does not require assumptions on spiked structure or bounded condition number. Leveraging variational inference and a novel local random matrix theory, our method constructs an adaptive, eigenvector-preserving shrinkage estimator capable of accommodating arbitrary spectral shapes and broad rank regimes. Unlike conventional linear shrinkage and existing nonlinear approaches, our estimator is theoretically grounded and provably enhances detection power for mean vector testing. Extensive simulations and real-data analyses—from finance to genomics—demonstrate its robustness and superior performance over state-of-the-art alternatives.
📝 Abstract
In this paper we study the problem of comparing the means of a single observation and a reference sample in the presence of a common data covariance matrix, where the data dimension $p$ grows linearly with the number of samples $n$ and $p/n$ converges to a number between 0 and 1. The approach we take is to replace the sample covariance matrix with a nonlinear shrinkage estimator -- i.e., a matrix with the same eigenvectors -- in Hotelling's $T^2$ test. Current approaches of this sort typically assume that the data covariance matrix has a condition number or spiked rank that increases slowly with dimension. However, this assumption is ill-suited to data sets containing many strongly correlated background covariates, as often found in finance, genetics, and remote sensing. To address this problem we construct, using variational methods and new local random-matrix laws, a nonlinear covariance shrinkage method tailored to optimize detection performance across a broad range of spiked ranks and condition numbers. We then demonstrate, via both simulated and real-world data, that our method outperforms existing approaches.