  This article inspects whether a multivariate distribution is different from a
specified distribution or not, and it also tests the equality of two
multivariate distributions. In the course of this study, a graphical tool-kit
using well-known half-spaced depth based information criteria is proposed,
which is a two-dimensional plot, regardless of the dimension of the data, and
it is even useful in comparing high-dimensional distributions. The simple
interpretability of the proposed graphical tool-kit motivates us to formulate
test statistics to carry out the corresponding testing of hypothesis problems.
It is established that the proposed tests based on the same information
criteria are consistent, and moreover, the asymptotic distributions of the test
statistics under contiguous/local alternatives are derived, which enable us to
compute the asymptotic power of these tests. Furthermore, it is observed that
the computations associated with the proposed tests are unburdensome. Besides,
these tests perform better than many other tests available in the literature
when data are generated from various distributions such as heavy tailed
distributions, which indicates that the proposed methodology is robust as well.
Finally, the usefulness of the proposed graphical tool-kit and tests is shown
on two benchmark real data sets.
