A nonparametric test of independence based on L_1-error

We propose a test of mutual independence between random vectors with arbitrary dimensions. Our approach is based on the $L_1-$distance between the joint density and the product of the marginal densities. We establish the asymptotic normal approximation of the corresponding statistic under the null hypothesis without assuming any regularity conditions. From a practical point of view, we perform numerical studies in order to assess the efficiency of our procedure and compare it to existing independence tests in the literature. For many examples investigated, the proposed test provides good performance compared with existing methods.