  Hotelling's $T^2$ test is a classical approach for discriminating 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 estimated 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.
