  Statistical analysis of tensor-valued data has largely used the
tensor-variate normal (TVN) distribution that may be inadequate when data comes
from distributions with heavier or lighter tails. We study a general family of
elliptically contoured (EC) tensor-variate distributions and derive its
characterizations, moments, marginal and conditional distributions, and the EC
Wishart distribution. We describe procedures for maximum likelihood estimation
from data that are (1) uncorrelated draws from an EC distribution, (2) from a
scale mixture of the TVN distribution, and (3) from an underlying but unknown
EC distribution, where we extend Tyler's robust estimator. A detailed
simulation study highlights the benefits of choosing an EC distribution over
the TVN for heavier-tailed data. We develop tensor-variate classification rules
using discriminant analysis and EC errors and show that they better predict
cats and dogs from images in the Animal Faces-HQ dataset than the TVN-based
rules. A novel tensor-on-tensor regression and tensor-variate analysis of
variance (TANOVA) framework under EC errors is also demonstrated to better
characterize gender, age and ethnic origin than the usual TVN-based TANOVA in
the celebrated Labeled Faces of the Wild dataset.
