Reducing Multidimensional Two-Sample Data to One-Dimensional Interpoint Comparisons

The most popular technique for reducing the dimensionality in comparing two multidimensional samples of X ∼ F and Y ∼ G is to analyze distributions of interpoint comparisons based on a univariate function h (e.g. the interpoint distances). We provide a theoretical foundation for this technique, by showing that having both i) the equality of the distributions of within sample comparisons (h(X1, X2) =Lh(Y1, Y2)) and ii) the equality of these with the distribution of between sample comparisons ((h(X1, X2) =Lh(X3, Y3)) is equivalent to the equality of the multivariate distributions (F = G).