A new normal reference test for linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA

In recent decades, with rapid development of data collecting technologies, high-dimensional data become increasingly prevalent, and much work has been done for hypotheses on mean vectors for high-dimensional data. However, only a few methods have been proposed and studied for the general linear hypothesis testing (GLHT) problem for high-dimensional data which includes many well-studied problems as special cases. A centralized L2-norm based test statistic is proposed and studied for the high-dimensional GLHT problem. It is shown that under some mild conditions, the proposed test statistic and a chi-square-mixture have the same normal or non-normal limiting distributions. It is then justified that the null distribution of the test statistic can be approximated by using that of the chi-square-type mixture. The distribution of the chi-square-type mixture can be well approximated by a three-cumulant matched chi-square approximation with its approximation parameters consistently estimated from the data. Since the chi-square-type mixture is obtained from the test statistic under the null hypothesis and when the data are normally distributed, the resulting test is termed as a normal reference test. The asymptotic power of the proposed test under a local alternative is established. The impact of the data non-normality on the proposed test is also studied. Three simulation studies and a real data example demonstrate that in terms of size control, the proposed test performs well regardless of whether the data are nearly uncorrelated, moderately correlated, or highly correlated and it outperforms two existing competitors substantially.