Inspecting discrepancy between multivariate distributions using half-space depth based information criteria

TL;DR

This paper introduces a graphical tool-kit and statistical tests based on half-space depth for comparing multivariate distributions, demonstrating robustness, interpretability, and superior performance on various data types including high-dimensional data.

Contribution

It proposes a novel graphical tool-kit and hypothesis tests using half-space depth criteria that are simple, interpretable, and effective for multivariate distribution comparison.

Findings

The tests are consistent and have known asymptotic distributions.

The proposed methods outperform existing tests on heavy-tailed distributions.

The graphical tool-kit is useful for visual comparison of distributions.

Abstract

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-space 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.