A robust and powerful metric for distributional homogeneity

Assessing the homogeneity of two random vectors is a fundamental task in statistical inference. In this work, we introduce a weighted multivariate Cramér‐von Mises type metric that transforms each variable through a marginal mixture distribution function and integrates the squared difference in probability functions of these transformed variables. Notably, our metric exhibits scale invariance, rendering it robust against outliers and heterogeneity. The expression for our metric is straightforward and possesses a closed‐form representation. It is non‐negative and attains a value of zero if and only if the two random vectors are identically distributed. Moreover, our metric employs an l1$$ {l}_1 $$‐norm expression, which significantly enhances its effectiveness in high‐dimensional scenarios compared to traditional methods relying on the l2$$ {l}_2 $$‐norm. We validate the efficacy of our proposed approach through extensive simulation studies and empirical data analysis.