Multivariate tests for the multi-sample location problem based on depth function

In this paper, a class of affine-invariant tests is presented for the multi-sample multivariate location problem. In the procedure to derive the asymptotic distribution of the tests under the null hypothesis, we do not require any symmetric assumption of the distribution functions. The asymptotic relative efficiency of the tests is discussed under the class of elliptically symmetric distributions. Further comparisons are made among several statistics using Monte Carlo results. Asymptotic relative efficiencies along with Monte Carlo results indicate that selected members of the proposed class perform very well for a broad class of distributions. Finally, we apply our proposed tests to Egyptian skulls data for multivariate five different periods comparisons.