Improved Test for High-Dimensional Mean Vectors and Covariance Matrices Using Random Projection

This paper proposes an improved random projection-based method for testing high-dimensional two-sample mean vectors and covariance matrices. For mean testing, the proposed approach incorporates training data to guide the construction of projection matrices toward the estimated mean difference, thereby substantially enhancing the power of the projected Hotelling’s T2 statistic. We introduce three aggregation strategies—maximum, average, and percentile-based—to ensure stable performance across multiple projections. For covariance testing, the method employs data-driven projections aligned with the leading eigenvector of the sample covariance matrix to amplify the differences between matrices. Aggregation strategies—maximum-, average-, and percentile-based for the mean problem and minimum and average p-values for the covariance problem—are developed to further stabilize performance across repeated projections. An application to gene expression data is provided to illustrate the method. Extensive simulation studies show that the proposed method performs favorably compared to a recent state-of-the-art technique, particularly in detecting sparse signals, while maintaining control of the Type-I error rate.