On summed nonparametric dependence measures in high dimensions, fixed or large samples

For the mutual independence testing problem, the use of summed nonparametric dependence measures, including Hoeffding's D, Blum-Kiefer-Rosenblatt's R, Bergsma-Dassios-Yanagimoto's ${\tau }^{⁎}$, is considered. The asymptotic normality of this class of test statistics for the null hypothesis is established when (i) both the dimension and the sample size go to infinity simultaneously, and (ii) the dimension tends to infinity but the sample size is fixed. The new result for the asymptotic regime (ii) is applicable to the HDLSS (High Dimension, Low Sample Size) data. Further, the asymptotic Pitman efficiencies of the family of considered tests are investigated with respect to two important sum-of-squares tests for the asymptotic regime (i): the distance covariance based test and the product-moment covariance based test. Formulae for asymptotic relative efficiencies are found. An interesting finding reveals that even if the population follows a normally distributed structure, the two state-of-art tests suffer from power loss if some components of the underlying data have different scales. Simulations are conducted to confirm our asymptotic results. A real data analysis is performed to illustrate the considered methods.