A high-dimensional robust test for distribution homogeneity under the new measure

Two sample testing is a fundamental problem in statistical inference. In this work, we propose a novel testing method, the Cauchy-Characteristic Function Discrepancy (CCFD), which leverages the uniqueness of the characteristic function of random variables and the special properties of the standard Cauchy distribution. Unlike traditional methods, CCFD features a closed-form expression, robustness to outliers, and does not rely on moment or continuity assumptions. By introducing the <i>l</i>₁-norm, CCFD exhibits superior detection capability in high-dimensional settings while maintaining excellent computational efficiency with a complexity of 𝑂(<i>n</i>²). Extensive simulations demonstrate that CCFD outperforms existing methods, including energy statistics, maximum mean discrepancy, and projection-based methods, particularly in scenarios with heavy-tailed distributions or equal moments. Additionally, a real-world application on the Land Mines dataset underscores the practical utility of CCFD. These findings establish CCFD as a robust, efficient, and broadly applicable tool for detecting distributional differences in modern data analysis.