Abstract

In the past two decades, much attention has been paid for high-dimensional hypothesis testing. Several centralized or non-centralized L2-norm based test statistics have been proposed. Most of them imposed strong assumptions on the underlying covariance structure of the high-dimensional data so that the associated test statistics are asymptotically normally distributed. In real data analysis, however, these assumptions are hardly checked so that the resulting tests have a size control problem when the required assumptions are not satisfied. To overcome this difficulty, in this talk, we investigate a so-called normal-reference test which can control the size well. In the normal-reference test, the null distribution of a test statistic is approximated with that of a chi-square-type mixture which is obtained from the test statistic when the null hypothesis holds and when the samples are normally distributed. The distribution of the chi-square-type mixture can be well approximated by a three-cumulant matched χ²-approximation with the approximation parameters consistently estimated from the data. Two simulation studies demonstrate that in terms of size control, the proposed normal-reference test performs well regardless of whether the data are nearly uncorrelated, moderately correlated, or highly correlated and it performs much better than two existing competitors. A real data example illustrates the proposed normal-reference test.