Large-Scale Multiple Testing for Matrix-Valued Data under Double Dependency

High-dimensional inference based on matrix-valued data has drawn increasing attention in modern statistical research, yet not much progress has been made in large-scale multiple testing specifically designed for analysing such data sets. Motivated by this, we consider in this article an electroencephalography (EEG) experiment that produces matrix-valued data and presents a scope of developing novel matrix-valued data based multiple testing methods controlling false discoveries for hypotheses that are of importance in such an experiment. The row-column cross-dependency of observations appearing in a matrix form, referred to as double-dependency, is one of the main challenges in the development of such methods. We address it by assuming matrix normal distribution for the observations at each of the independent matrix data-points. This allows us to fully capture the underlying double-dependency informed through the row- and column-covariance matrices and develop methods that are potentially more powerful than the corresponding one (e.g., Fan and Han (2017)) obtained by vectorizing each data point and thus ignoring the double-dependency. We propose two methods to approximate the false discovery proportion with statistical accuracy. While one of these methods is a general approach under double-dependency, the other one provides more computational efficiency for higher dimensionality. Extensive numerical studies illustrate the superior performance of the proposed methods over the principal factor approximation method of Fan and Han (2017). The proposed methods have been further applied to the aforementioned EEG data.