  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.
