Some multivariate goodness of fit tests based on data depth

TL;DR

This paper introduces new multivariate goodness of fit tests utilizing data depth functions, which are distribution-free and based on univariate tests, with demonstrated finite sample properties and real data application.

Contribution

It develops novel multivariate goodness of fit tests using data depth, addressing computational challenges and extending univariate tests to multivariate settings.

Findings

Test statistics closely approximate true depth with large samples.

Proposed tests are distribution-free under the null hypothesis.

Finite sample properties are validated through numerical examples.

Abstract

Using the fact that some depth functions characterize certain family of distribution functions, and under some mild conditions, distribution of the depth is continuous, we have constructed several new multivariate goodness of fit tests based on existing univariate GoF tests. Since exact computation of depth is difficult, depth is computed with respect to a large random sample drawn from the null distribution. It has been shown that test statistic based on estimated depth is close to that based on true depth for a large random sample from the null distribution. Some two sample tests for scale difference, based on data depth are also discussed. These tests are distribution-free under the null hypothesis. Finite sample properties of the tests are studied through several numerical examples. A real data example is discussed to illustrate usefulness of the proposed tests.