Mean Test for Matrix-Valued Data by Optimal Projection

Matrix-valued data have become increasingly prevalent in modern applications, and hypothesis testing on the mean matrix stands as a fundamental problem that can guide downstream statistical analyses. In this paper, we propose a projection-based test for the mean matrix, which incorporates the inherent structural information of matrix-valued data. Based on the estimators of mean and covariance matrices, our method estimates the optimal projection direction through a constrained and regularized quadratic optimization problem. Theoretically, we establish the entry-wise error bound for the low-rank approximation of the mean matrix under correlated noise, and further derive the consistency of the estimated projection direction when the sample size diverges or remains fixed while the data dimensions grow large. These results ensure the validity and power of the proposed test. In addition, we develop a computationally efficient testing procedure specifically designed for the rank-one mean matrix scenario. Both simulation studies and the real data analysis demonstrate the effectiveness of the proposed tests. Supplementary materials for this article are available online.