An Adaptive Test Of The Independence Of High-dimensional Data Based On Kendall Rank Correlation Coefficient

With the development of modern technology and social economy,the demand for high-dimensional data analysis in various fields is increasing.One of the characteristics of high-dimensional data is that both the sample size n and the sample dimension p tend to infinity.However,most traditional statistical methods generally assume that the sample size n is large and the sample dimension p is relatively fixed.These research methods have poor or even invalid testing results for high-dimensional data.Therefore,it is one of the hot issues in modern statistics to explore how to handle statistical inference problems from high-dimensional data.This paper presents an adaptive test for the independence of high-dimensional continuous data based on the Kendall rank correlation coefficient.Firstly,under the null assumption that the components of the random vector are mutually independent,we establish test statistics based on the Kendall rank correlation coefficient;Secondly,we calculate the partial higher-order origin moments of the Kendall rank correlation coefficient,and then calculate the mean,variance,and covariance of the test statistics.Using the central limit theorem related to martingale difference,it is proved that the test statistic asymptotically obey normal distribution;Finally,based on the conclusion of He et al.(2021)about the asymptotic properties of U statistics,it is proved that the test statistics proposed in this paper are asymptotically independent of the maximum statistic in Han & Liu(2014).Based on the above theory,this paper proposes an adaptive test method for independence testing of high-dimensional continuous data.For sparse and non-sparse data,the proposed adaptive testing method can significantly enhance power without distortion of size.With the help of R software,we conduct numerical simulation studies under different continuous population distributions,the effectiveness of the adaptive testing method proposed in this paper is verified by analyzing the two indicators of size and power.