We propose a novel independence measure for two random vectors, termed the Adaptive Hellinger Distance (AHD). This measure employs an innovative variant of the Hellinger distance (HD) and integrates methodological insights from the HHG test (introduced in Heller et al. (2012)). The proposed AHD test can be regarded as a crucial extension of the HHG test, as it shares a similar analytical framework while addressing certain limitations. Specifically, like the HHG test, it can detect dependence between two random vectors in any dimension, regardless of whether they are continuous or discrete. The rank test based on AHD performs exceptionally well and demonstrates robustness in applications. Furthermore, we have developed two tests utilizing p-value combination methods, named AHD-Simes and AHD-CCT. These methods are designed to integrate the results of tests conducted on four bins, ultimately enhancing overall performance. Simulation studies and real data analysis simultaneously illustrate the power and the utility of the proposed method, which is highly competitive especially when dealing with random vectors that are heavy-tailed or have outliers. We applied the proposed method to analyze two real datasets with different data types. Additionally, a new R package named AHD is developed to implement the proposed method.
