Some multivariate goodness of fit tests based on data depth

Abstract

Using notions of depth functions in the multivariate setting, we have constructed several new multivariate goodness of fit (GoF) tests based on existing univariate GoF tests. Since the exact computation of depth is difficult, depth is estimated based on a large random sample drawn from the null distribution. It has been shown that test statistics based on estimated depth are close to those based on the true depth. Some two-sample tests based on data depth are also discussed for scale differences. These tests are distribution-free under the null hypothesis. Finite sample properties of the proposed tests are studied using several numerical examples. A real-data example is discussed to illustrate the usefulness of the proposed GoF tests.