  Modern datasets are often in the form of matrices or arrays,potentially
having correlations along each set of data indices. For example, data involving
repeated measurements of several variables over time may exhibit temporal
correlation as well as correlation among the variables. A possible model for
matrix-valued data is the class of matrix normal distributions, which is
parametrized by two covariance matrices, one for each index set of the data. In
this article we describe an extension of the matrix normal model to accommodate
multidimensional data arrays, or tensors. We generate a class of array normal
distributions by applying a group of multilinear transformations to an array of
independent standard normal random variables. The covariance structures of the
resulting class take the form of outer products of dimension-specific
covariance matrices. We derive some properties of these covariance structures
and the corresponding array normal distributions, discuss maximum likelihood
and Bayesian estimation of covariance parameters and illustrate the model in an
analysis of multivariate longitudinal network data.
