Using space filling curves to compare two multivariate distributions with distribution-free tests

This paper proposes new tests to compare two multivariate probability distributions. Since basic ranks do not canonically exist in Rd, it is impossible to have a natural multivariate generalisation of rank-based tests such as the two-sample Kolmogorov–Smirnov test. We thus rely on recent measure transportation theory to transform this d−dimensional problem into one-dimensional classical tests by using space filling curves. To foster lower computation time, we develop distribution-free tests so as to avoid computing critical values for any particular problem. We demonstrate their theoretical validity and compare them to each other and to the existing distribution-free techniques via extensive simulations. We show that they are computationally efficient and most of the time outperform existing techniques when measuring the corresponding power functions. Finally, we apply the proposed tests to a dataset about stars’ luminosity and temperature.