With the rise of the "big data" phenomenon in recent years, data is coming in
many different complex forms. One example of this is multi-way data that come
in the form of higher-order tensors such as coloured images and movie clips.
Although there has been a recent rise in models for looking at the simple case
of three-way data in the form of matrices, there is a relative paucity of
higher-order tensor variate methods. The most common tensor distribution in the
literature is the tensor variate normal distribution; however, its use can be
problematic if the data exhibit skewness or outliers. Herein, we develop four
skewed tensor variate distributions which to our knowledge are the first skewed
tensor distributions to be proposed in the literature, and are able to
parameterize both skewness and tail weight. Properties and parameter estimation
are discussed, and real and simulated data are used for illustration.