In this talk, we consider a general linear hypothesis testing (GLHT) problem in high-dimensional linear regression where the data dimension may be much larger than the sample size. Recently, several non-scale and scale-invariant tests have been proposed for this important GLHT problem which includes one-way and two-way MANOVA tests as special cases. Most of these tests impose strong assumptions on the underlying covariance matrix so that their test statistics are asymptotically normally distributed. However, a simulation example and some theoretical justifications indicate that these assumptions may not be satisfied or hardly be checked so that these tests may not be able to maintain the nominal size well in practice. Some scale-invariant tests employ a so-called adjustment coefficient to improve the convergence of their test statistics to normal distributions. However, it turns out that this adjustment coefficient works well in improving the convergence of a scale-invariant test to a normal distribution when the underlying null distribution of the test is approximately normal but it substantially worsens the size control and power of the test otherwise. To overcome these problems, in this talk, we propose a simple and adaptive scale-invariant test which has good size control and power without imposing strong assumptions on the underlying covariance or correlation matrix. In addition, we adopt a strategy to employ the adjustment coefficient smartly to the proposed test. Simulation and real data examples demonstrate the good performance of the proposed test in terms of size control and power, via comparing it against several non-scale and scale-invariant tests.
Joint work with Liang Zhang, Tianming Zhu, and Jin-Ting Zhang.