  We are interested in testing general linear hypotheses in a high-dimensional
multivariate linear regression model. The framework includes many well-studied
problems such as two-sample tests for equality of population means, MANOVA and
others as special cases. A family of rotation-invariant tests is proposed that
involves a flexible spectral shrinkage scheme applied to the sample error
covariance matrix. The asymptotic normality of the test statistic under the
null hypothesis is derived in the setting where dimensionality is comparable to
sample sizes, assuming the existence of certain moments for the observations.
The asymptotic power of the proposed test is studied under various local
alternatives. The power characteristics are then utilized to propose a
data-driven selection of the spectral shrinkage function. As an illustration of
the general theory, we construct a family of tests involving ridge-type
regularization and suggest possible extensions to more complex regularizers. A
simulation study is carried out to examine the numerical performance of the
proposed tests.
