  It has been proposed that complex populations, such as those that arise in
genomics studies, may exhibit dependencies among observations as well as among
variables. This gives rise to the challenging problem of analyzing unreplicated
high-dimensional data with unknown mean and dependence structures.
Matrix-variate approaches that impose various forms of (inverse) covariance
sparsity allow flexible dependence structures to be estimated, but cannot
directly be applied when the mean and covariance matrices are estimated
jointly. We present a practical method utilizing generalized least squares and
penalized (inverse) covariance estimation to address this challenge. We
establish consistency and obtain rates of convergence for estimating the mean
parameters and covariance matrices. The advantages of our approaches are: (i)
dependence graphs and covariance structures can be estimated in the presence of
unknown mean structure, (ii) the mean structure becomes more efficiently
estimated when accounting for the dependence structure among observations; and
(iii) inferences about the mean parameters become correctly calibrated. We use
simulation studies and analysis of genomic data from a twin study of ulcerative
colitis to illustrate the statistical convergence and the performance of our
methods in practical settings. Several lines of evidence show that the test
statistics for differential gene expression produced by our methods are
correctly calibrated and improve power over conventional methods.
