A New Depth-Based Test for Multivariate Two-Sample Problems

Statistical depth provides a center–outward ordering of multivariate observations and is widely used in nonparametric inference. We study depth-based tests for multivariate two-sample problems and examine the behaviour of different depth notions using the DD plot (data-depth plot) across a variety of distributional space. The DD plot illustrates that depth functions differ in their sensitivity to distributional differences, emphasizing the importance of depth selection in two-sample testing. We propose a new two-sample test statistic, log DDR, constructed from ratios of numerical depth values rather than depth-induced ranks. Simulation studies under multiple scenarios and for three representative depth functions indicate that log DDR achieves improved power relative to several competing depth-based nonparametric tests. The results further demonstrate that the performance of log DDR and existing methods depends strongly on the chosen depth function, consistent with insights from the DD plot. These findings support a two-stage testing approach in which the DD plot is used to guide the choice of depth notion before applying log DDR for homogeneity testing.