An adaptive test of the independence of high-dimensional data based on Kendall rank correlation coefficient

The paper considers the test on the components independence of high-dimensional continuous random vectors based on the Kendall rank correlation coefficient. Under the null hypothesis, it proves that the sum-of-powers type statistics constructed by the Kendall rank correlation coefficient are asymptotically normally distributed and independent when the sum of the orders is odd, and they are also independent with the extreme-value-type test statistics of the Kendall rank correlation coefficient. Based on the asymptotic independence property, it proposes an adaptive test procedure which combines p-values from the sum-of-powers-type test and extreme-value-type test. Some simulations reveal that the proposed test method can significantly enhance the statistical power of the test while keep the type I error well controlled, and it is powerful against a wide range of alternatives (including dense, sparse or moderately sparse signals). To illustrate the use of the proposed adaptive test method, two real data sets are also analysed.