Independence tests for continuous random variables based on the longest increasing subsequence

We propose a new class of nonparametric tests for the supposition of independence between two continuous random variables X and Y . Given a size n sample, let π be the permutation which maps the ranks of the X observations on the ranks of the Y observations. We identify the independence assumption of the null hypothesis with the uniform distribution on the permutation space. A test based on the size of the longest increasing subsequence of   π ( L n ) is defined. The exact distribution of L n is computed from Schensted’s theorem (Schensted, 1961). The asymptotic distribution of L n was obtained by Baik et al. (1999). As the statistic L n is discrete, there is a small set of possible significance levels. To solve this problem we define the J L n statistic which is a jackknife version of L n , as well as the corresponding hypothesis test. A third test is defined based on the J L M n statistic which is a jackknife version of the longest monotonic subsequence of π . On a simulation study we apply our tests to diverse dependence situations with null or very small correlations where the independence hypothesis is difficult to reject. We show that L n , J L n and J L M n tests have very good performance on that kind of situations. We illustrate the use of those tests on two real data examples with small sample size.