An overview of tests on high-dimensional means

Testing high-dimensional means has many applications in scientific research. For instance, it is of great interest to test whether there is a difference of gene expressions between control and treatment groups in genetic studies. This can be formulated as a two-sample mean testing problem. However, the Hotelling T² test statistic for the two-sample mean problem is no longer well defined due to singularity of the sample covariance matrix when the sample size is less than the dimension of data. Over the last two decades, the high-dimensional mean testing problem has received considerable attentions in the literature. This paper provides a selective overview of existing testing procedures in the literature. We focus on the motivation of the testing procedures, the insights into how to construct the test statistics and the connections, and comparisons of different methods.