On high-dimensional tests for mutual independence based on Pearson's correlation coefficient

This paper investigates a class of statistics based on Pearson's correlation coefficient for testing the mutual independence of a random vector in high dimensions. Two existing statistics, proposed by Schott (2005) and Mao (2014) respectively, are special cases of the class. A generic testing theory for the class of statistics is developed, which clarifies under what conditions the class of statistics can be employed for the testing purpose. By virtue of the theory, three new tests are introduced, and related statistical properties are discussed. To examine our theoretical findings and check the performance of the new tests, simulation studies are applied. The simulation results justify the theoretical findings and show that the newly introduced tests perform well, as long as both the dimension and the sample size of the data are moderately large.