Max-sum test based on Spearman's footrule for high-dimensional independence tests

Testing high-dimensional data independence is an essential task of multivariate data analysis in many fields. Typically, the quadratic and extreme value type statistics based on the Pearson correlation coefficient are designed to test dense and sparse alternatives for evaluating high-dimensional independence. However, the two existing popular test methods are sensitive to outliers and are invalid for heavy-tailed error distributions. To overcome these problems, two test statistics, a Spearman's footrule rank-based quadratic scheme and an extreme value type test for dense and sparse alternatives, are proposed, respectively. Under mild conditions, the large sample properties of the resulting test methods are established. Furthermore, the proposed two test statistics are proved to be asymptotically independent. The max-sum test based on Spearman's footrule statistic is developed by combining the proposed quadratic with extreme value statistics, and the asymptotic distribution of the resulting statistical test is established. The simulation results demonstrate that the proposed max-sum test performs well in empirical power and robustness, regardless of whether the data is sparse dependence or not. Finally, to illustrate the use of the proposed test method, two empirical examples of Leaf and Parkinson's disease datasets are provided.