Unified Ball Distance: a metric on probability measures with applications to nonparametric two-sample testing

Ball Divergence (BD) is a robust, ball-based measure of distributional dissimilarity, but it is not a metric and therefore lacks a coherent geometric scale for distance maps, clustering, and multidimensional scaling. We introduce the Unified Ball Distance (UBD), which retains BD's equality-zero property while endowing probability measures with metric geometry. UBD anchors ball comparisons to a shared reference design, and we prove that its square root satisfies the metric axioms. The sample UBD yields a simple nonparametric two-sample test with a distance scale comparable across pairs. We establish almost-sure consistency, finite-sample concentration inequalities, and the null asymptotic distribution. Simulations show competitive and often higher power under heavy tails and scale changes. A three-arm antidepressant study shows that the UBD distance matrix matches the clinical efficacy ordering, enabling geometry-driven workflows such as clustering and multidimensional scaling.