Semi-Distance Correlation and Its Applications

We propose a new measure of dependence between a categorical random variable and a random vector with potentially high dimensions, named semi-distance correlation. It is an interesting extension of distance correlation to accommodate the information of the categorical random variable. It equals zero if and only if the categorical random variable and the other random vector are independent. Two important applications of semi-distance correlation are considered. First, we develop a semi-distance independence test between a categorical random variable and a random vector and derive its asymptotic distributions. When the dimension of the random vector tends to infinity, we derive the explicit asymptotic normal distribution of the test statistic under the null hypothesis, which allows us to compute p-values in an efficient and fast way for high dimensional data. Second, we propose to use the semi-distance correlation as a marginal utility between the response and a group of covariates to do groupwise variable screening for ultrahigh dimensional classification problems. The sure screening property has also been established. Monte Carlo simulations and a real data application are presented to demonstrate the excellent finite sample property of the proposed procedures. A new R package semidist is also developed to implement the proposed methods. Supplementary materials for this article are available online.