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<p>We treat the problem of measuring and testing for the equality of multivariate distributions. We construct a new measure by combining the concept of projection averaging with the theory of optimal transport. Our measure possesses the necessary and intuitive property as a metric index. Briefly, it is zero almost surely if and only if two multivariate random variables have the same distribution. The sample counterparts of this measure can be expressed elegantly and enjoy desirable theoretical properties. Based on the corresponding U- and V-statistics, we propose two nonparametric tests for assessing the equality of two multivariate probability distributions. The proposed tests are free of underlying population distributions and are consistent against all fixed alternatives. Moreover, our procedures avoid the need to estimate any nuisance parameters or impose moment assumptions, which broadens the scope of the proposed testing applications and reduces the computational burden. We also demonstrate the favorable finite-sample performance of the proposed tests through extensive simulations and real data examples, particularly under certain models involving location differences.</p>

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A fully distribution-free and consistent two-sample test based on projection averaging

  • 期刊论文

  • 发表日期：

    2026年5月23日

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  • Huijun Shi,
  • Daojiang He,
  • Kai Xu

摘要

We treat the problem of measuring and testing for the equality of multivariate distributions. We construct a new measure by combining the concept of projection averaging with the theory of optimal transport. Our measure possesses the necessary and intuitive property as a metric index. Briefly, it is zero almost surely if and only if two multivariate random variables have the same distribution. The sample counterparts of this measure can be expressed elegantly and enjoy desirable theoretical properties. Based on the corresponding U- and V-statistics, we propose two nonparametric tests for assessing the equality of two multivariate probability distributions. The proposed tests are free of underlying population distributions and are consistent against all fixed alternatives. Moreover, our procedures avoid the need to estimate any nuisance parameters or impose moment assumptions, which broadens the scope of the proposed testing applications and reduces the computational burden. We also demonstrate the favorable finite-sample performance of the proposed tests through extensive simulations and real data examples, particularly under certain models involving location differences.