In a highly influential paper, Bai and Saranadasa (1996) proposed a test for determining equality of means of two normal populations when the population variances are assumed to be equal. The classical test for this problem, Hotelling’s $T^2$ test, loses power when the dimension $p$ grows with the sample size $n$, and becomes ineffective when $p/n$ approaches 1. In contrast, the test by Bai and Sarandasa has nontrivial power against a wide range of alternatives as long as the ratio $p/n$ remains bounded. The test was modified by Chen and Qin (2010) to extend the scope to the setting where $p$ can be an order of magnitude larger than $n$.
In this talk, we review two recently proposed regularized test procedures for the two sample testing problem in the $p/n to c in (0,infty)$ setting. The first one is based on ridge regularization of the sample covariance matrix, and the second one is based on reweighing the eigen-subspaces of the sample covariance matrix under the assumption of a spiked population covariance. We consider a random family of alternatives determined by the unknown population covariance matrix, an under this Bayesian framework, we find the optimal regularization parameter. This framework also enables us show that the test by Bai and Saranadasa is minimax under appropriate structural constraints, for both spiked and non-spiked settings.
(This talk is based on joint work with Haoran Li, Alexander Aue, Jie Peng and Pei Wang)