A Multivariate Two-Sample Location Test Based on Empirical Distribution Functions

We propose a novel multivariate two-sample location test based on the empirical distribution function (EDF). The test statistic is constructed using a power divergence measure between the EDFs of two independent samples, enhancing sensitivity under both heavy-tailed and light-tailed distributions. Implemented via the permutation principle, the test offers flexibility and robustness across a wide range of data conditions. A notable feature is its component-wise scale invariance, eliminating the need to standardize variables. We derive the asymptotic distribution of the test statistic under the null hypothesis and general conditions, showing that when sample sizes are equal, it converges to a linear combination of independent chi-square variables with two degrees of freedom. The asymptotic distribution of the test statistic is independent of data dimensionality, offering a significant advantage in high-dimensional settings. Unlike Hotelling’s T$${}^{2\ }$$and related methods, our approach does not require estimating a dispersion matrix, thereby avoiding issues such as singularity or instability in small samples. We also provide explicit expressions for the expectation and variance of the test statistic under the null hypothesis. Monte Carlo simulations demonstrate that the proposed test consistently achieves higher empirical power than existing alternatives under a wide range of distributional shapes, including heavy-tailed, light-tailed, and elliptically asymmetric populations. To illustrate its practical utility, we apply the method to the Peterson dataset from Van Belle et al. (2004), which explores the association between sudden infant death syndrome (SIDS) and birth weight. The results highlight the effectiveness and robustness of the proposed test in real-world applications, establishing it as a powerful and versatile alternative to traditional multivariate methods. Overall, the proposed test consistently delivers better statistical power across a wide range of population characteristics, making it a robust and versatile alternative to existing methods.