A distribution-free test of independence based on robust semi-Gaussian kernel distance covariance

In this paper, we propose a novel robust semi-Gaussian kernel distance (RSGKD) covariance to measure dependence between a random vector X and a categorical random variable Y. Our proposed RSGKD covariance satisfies the independence-zero equivalence property, i.e. the RSGKD covariance is non-negative and equals zero if and only if X and Y is statistically independent. Since our proposed RSGKD covariance does not require moment restrictions on random vectors, it is robust to heavy-tailed distributions or outliers in the dataset. Under some conditions, the asymptotic properties of our proposed test statistic are established. Furthermore, both extensive numerical simulations and a real data analysis demonstrate the satisfactory finite-sample performance of our proposed test.