Symmetrical Independence Tests for Two Random Vectors with Arbitrary Dimensional Graphs

Test of independence between random vectors X and Y is an essential task in statistical inference. One type of testing methods is based on the minimal spanning tree of variables X and Y . The main idea is to generate the minimal spanning tree for one random vector X , and for each edges in minimal spanning tree, the corresponding rank number can be calculated based on another random vector Y . The resulting test statistics are constructed by these rank numbers. However, the existed statistics are not symmetrical tests about the random vectors X and Y such that the power performance from minimal spanning tree of X is not the same as that from minimal spanning tree of Y . In addition, the conclusion from minimal spanning tree of X might conflict with that from minimal spanning tree of Y . In order to solve these problems, we propose several symmetrical independence tests for X and Y. The exact distributions of test statistics are investigated when the sample size is small. Also, we study the asymptotic properties of the statistics. A permutation method is introduced for getting critical values of the statistics. Compared with the existing methods, our proposed methods are more efficient demonstrated by numerical analysis.