Inspecting differences between multivariate distributions: graphical tool-kit and related tests

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 based on well-known half-spaced depth 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 are consistent, and moreover, the asymptotic distributions of the test statistics under contiguous 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.