In statistics, one of the fundamental inferential problems is to test a general linear hypothesis of regression coefficients under a linear model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. The testing problem is well-studied when the sample size is much larger than the dimension, but remains underexplored under high dimensional settings. Various classical invariant tests, despite their popularity in multivariate analysis, involve the inverse of the residual covariance matrix, which is inconsistent or even singular when the dimension is at least comparable to the degree of freedom. Consequently, classical tests perform poorly.
In this talk, I seek to regularize the spectrum of the residual covariance matrix by flexible shrinkage functions. A family of rotation-invariant tests is proposed. The asymptotic normality of the test statistics under the null hypothesis is derived in the setting where dimensionality is comparable to the sample size. The asymptotic power of the proposed test is studied under a class of 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, a family of tests involving ridge-type regularization is constructed.