On the properties of distance covariance for categorical data: Robustness, sure screening, and approximate null distributions

Pearson's Chi-squared test, though widely used for detecting association between categorical variables, exhibits low statistical power in large sparse contingency tables. To address this limitation, two novel permutation tests have been recently developed: the distance covariance permutation test and the U-statistic permutation test. Both leverage the distance covariance functional but employ different estimators. In this work, we explore key statistical properties of the distance covariance for categorical variables. Firstly, we show that unlike Chi-squared, the distance covariance functional is B-robust for any number of categories (fixed or diverging). Second, we establish the strong consistency of distance covariance screening under mild conditions, and simulations confirm its advantage over Chi-squared screening, especially for large sparse tables. Finally, we derive an approximate null distribution for a bias-corrected distance correlation estimate, demonstrating its effectiveness through simulations.