Unified Semi‐Distance‐Based Approaches for Equal Distribution Testing With Applications

With the advancement of data collection techniques, researchers frequently encounter complex data objects across various domains. One common interest lies in determining whether two groups of complex data objects originate from the same population. This paper introduces and examines a fast and accurate unified semi‐distance‐based approach designed to tackle this challenge. The approach exhibits broad applicability across a wide range of research areas, such as bioinformatics, audiology, environmentology, finance, and more. It effectively identifies differences between the distributions of two complex datasets, including both high‐dimensional data and functional data. The asymptotic null and alternative distributions of the proposed test statistic are established. Unlike the permutation approach, a unified, rapid and precise method to approximate the null distribution is proposed. Furthermore, the proposed test is shown to be root‐ consistent. Numerical results are presented for demonstrating the excellent performance of the proposed test in terms of size control, power, and computational cost. Additionally, the applications of the proposed test are showcased through examples involving both high‐dimensional data and functional data.