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A common disadvantage in existing distribution-free two-sample testingapproaches is that the computational complexity could be high. Specifically, ifthe sample size is $N$, the computational complexity of those two-sample testsis at least $O(N^2)$. In this paper, we develop an efficient algorithm withcomplexity $O(N \log N)$ for computing energy statistics in univariate cases.For multivariate cases, we introduce a two-sample test based on energystatistics and random projections, which enjoys the $O(K N \log N)$computational complexity, where $K$ is the number of random projections. Wename our method for multivariate cases as Randomly Projected Energy Statistics(RPES). We can show RPES achieves nearly the same test power with energystatistics both theoretically and empirically. Numerical experiments alsodemonstrate the efficiency of the proposed method over the competitors.

An Efficient and Distribution-Free Two-Sample Test Based on Energy Statistics and Random Projections