Differential Distance Correlation and Its Applications

In this paper, we propose a novel Euclidean-distance-based coefficient, named differential distance correlation, to measure the strength of dependence between a random variable $ Y \in \mathbb{R} $ and a random vector $ \boldsymbol{X} \in \mathbb{R}^p $. The coefficient has a concise expression and is invariant to arbitrary orthogonal transformations of the random vector. Moreover, the coefficient is a strongly consistent estimator of a simple and interpretable dependent measure, which is 0 if and only if $ \boldsymbol{X} $ and $ Y $ are independent and equal to 1 if and only if $ Y $ determines $ \boldsymbol{X} $ almost surely. An alternative approach is also proposed to address the limitation that the coefficient is non-robust to outliers. Furthermore, the coefficient exhibits asymptotic normality with a simple variance under the independent hypothesis, facilitating fast and accurate estimation of $ p $-value for testing independence. Three simulation experiments show that the proposed coefficient is more computationally efficient for independence testing and more effective in detecting oscillatory relationships than several competing methods. We also apply our method to analyze a real data example.