Abstract: The matrix-variate normal distribution is a popular model for high-dimensional transposable data because it decomposes the dependence structure of the random matrix into the Kronecker product of two covariance matrices, one for each of the row and column variables. However, few hypothesis testing procedures exist for these covariance matrices in high-dimensional settings. Therefore, we propose tests that assess the sphericity, identity, and diagonality hypotheses for the row (column) covariance matrix in a high-dimensional setting, while treating the column (row) dependence structure as a “nuisance” parameter. The proposed tests are robust to normality departures, provided that the Kronecker product dependence structure holds. In simulations, the proposed tests appear to maintain the nominal level, and tend to be powerful against the alternative hypotheses tested. The utility of the proposed tests is demonstrated by analyzing a microarray and an electroencephalogram study. The proposed testing methodology is implemented in the R package HDTD.

Key words and phrases: Covariance matrix, high-dimensional settings, hypothesis testing, matrix-valued random variables, transposable data.