A nonparametric independence test via penalized mutual information

In this paper, we propose a nonparametric independence test based on mutual information. Distinguished from the existing works, we estimate the mutual information in a conditional density form, whose dimension could be reduced to 1 with projection pursuit. The optimal projection direction is estimated by maximizing a penalized mutual information. Based on the optimal projection, we construct an independence test via the projected mutual information, which is insensitive to the dimensions of random vectors. The test is consistent against global alternatives, and can detect local alternatives at a fast rate as if the variables were univariate. Numerical results indicate that the test is more powerful compared with other existing independence tests, especially when the sample size is small or the dimension is large. We also apply the method to a stock portfolio performance data and show the superior performance of the new test.