Hypothesis Testing for the Covariance Matrix in High-Dimensional Transposable Data with Kronecker Product Dependence Structure

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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. We develop tests for assessing the form of the row (column) covariance matrix in high-dimensional settings while treating the column (row) dependence structure as a nuisance. Our tests are robust to normality departures provided that the Kronecker product dependence structure holds. In simulations, we observe that the proposed tests maintain the nominal level and are powerful against the alternative hypotheses tested. We illustrate the utility of our approach by examining whether genes associated with a given signalling network show correlated patterns of expression in different tissues and by studying correlation patterns within measurements of brain activity collected using electroencephalography.