An Improvement on the Hotelling $T^{2}$ Test Using the Ledoit-Wolf Nonlinear Shrinkage Estimator

Abstract

Hotelling's <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$T^{2}$</tex> test is a classical approach for dis-criminating the means of two multivariate normal samples that share a population covariance matrix. Hotelling's test is not ideal for high-dimensional samples because the eigenvalues of the esti-mated sample covariance matrix are inconsistent estimators for their population counterparts. We replace the sample covariance matrix with the nonlinear shrinkage estimator of Ledoit and Wolf 2020. We observe empirically for sub-Gaussian data that the resulting algorithm dominates past methods (Bai and Saranadasa 1996, Chen and Qin 2010, and Li et al. 2020) for a family of population covariance matrices that includes matrices with high or low condition number and many or few nontrivial-i.e., spiked-eigenvalues.