Overview of normal-reference tests for high-dimensional means with implementation in the R package ‘HDNRA’

The challenge of testing for equal mean vectors in high-dimensional data poses significant difficulties in statistical inference. Much of the existing literature introduces methods that often rely on stringent regularity conditions for the underlying covariance matrices, enabling asymptotic normality of test statistics. However, this can lead to complications in controlling test size. To address these issues, a new set of tests has emerged, leveraging the normal-reference approach to improve reliability. The latest normal-reference methods for testing equality of mean vectors in high-dimensional samples, potentially with differing covariance structures, are reviewed. The theoretical underpinnings of these tests are revisited, providing a new unified justification for the validity of centralized $L^2$-norm-based normal-reference tests (NRTs) by deriving the convergence rate of the distance between the null distribution of the test statistic and its corresponding normal-reference distribution. To facilitate practical application, an R package, HDNRA, is introduced, implementing these NRTs and extending beyond the two-sample problem to accommodate general linear hypothesis testing (GLHT). The package, designed with user-friendliness in mind, achieves efficient computation through a core implemented in C++ using Rcpp, OpenMP, and RcppArmadillo. Examples with real datasets are included, showcasing the application of various tests and providing insights into their practical utility.