Abstract: Recently, several non-scale-invariant and scale-invariant tests have been proposed for a general linear hypothesis testing problem for high-dimensional data, which include one-way and two-way MANOVA tests as special cases. Many of these tests impose strong assumptions on the underlying covariance matrix to ensure that their test statistics are asymptotically normally distributed. However, a simulation example and some theoretical justifications indicate that these assumptions are rarely satisfied in practice. As a result, these tests may not be able to maintain their nominal size well. To overcome this problem, we propose a normal-reference scale-invariant test. The test has good size control and power, without imposing strong assumptions on the underlying covariance or correlation matrix. A real-data example and several simulation studies demonstrate that the proposed test has much better size control and power than several non-scale-invariant and scale-invariant tests.

Key words and phrases: General linear hypothesis testing, high-dimensional linear regression, scale-invariant test.