On some characterizations and multidimensional criteria for testing homogeneity, symmetry and independence

We propose three new characterizations and corresponding distance-based weighted test criteria for the two-sample problem, and for testing symmetry and independence with multivariate data. All quantities have the common feature of involving characteristic functions, and it is seen that these quantities are intimately related to some earlier methods, thereby generalizing them. The connection rests on a special choice of the weight function involved. Equivalent expressions of the distances in terms of densities are given as well as a Bayesian interpretation of the weight function is involved. The asymptotic behavior of the tests is investigated both under the null hypothesis and under alternatives, and affine invariant versions of the test criteria are suggested. Numerical studies are conducted to examine the performances of the criteria. It is shown that the normal weight function, which is the hitherto most often used, is seriously suboptimal. The procedures are biased in the sense that the corresponding test criteria degenerate in high dimension and hence a bias correction is required as the dimension tends to infinity.