A non-parametric test of independence

We propose a new class of nonparametric tests for the supposition of independence between two continuous random variables X and Y: Given a sample of (X;Y ); the tests are based on the size of the longest increasing subsequence (from now on L.I.S.) of the permutation which maps the ranks of the X observations to the ranks of the Y observations. We identify the independence assumption between the two continuous variables with the space of permutation equipped with the uniform distribution and we show the exact distribution of the L.I.S. We calculate the distribution for several sample sizes. Through a simulation study we estimate the power of our tests for diverse alternative hypothesis under the null hypothesis of independence. We show that our tests have a remarkable behavior when the sample has very low correlation and visible (or hidden) dependence. We illustrate the use of our test on two real data examples with small sample sizes.