An adaptive test based on Kendall's tau for independence in high dimensions

We consider testing the mutual independency for high-dimensional data. It is known that $ L_2 $ L2-type statistics have lower power under sparse alternatives and $ L_\infty $ L∞-type statistics have lower power under dense alternatives in high dimensions. In this paper, we develop an adaptive test based on Kendall's tau to compromise both situations of the alternative, which can automatically be adapted to the underlying data. An adaptive test is very useful in practice as the sparsity or density for a data set is usually unknown. In addition, we establish the asymptotic joint distribution of $ L_2 $ L2-type and $ L_\infty $ L∞-type statistics based on Kendall's tau under mild assumptions and the asymptotic null distribution of the proposed statistic. Simulation studies show that our adaptive test performs well in either dense or sparse cases. To illustrate the usefulness and effectiveness of the proposed test, real data sets are also analysed.