Abstract

The graph of a bivariate sample defines a permutation. Statistics calculated on this permutation had been used to detect statistical dependence between pairs of random variables; see Garcia and Gonzalez-Lopez (2020) [1] and Garcia and Gonzalez-Lopez (2014) [2]. This work studies the relationship between statistical dependence and the shape of the Young tableau of the permutation defined from a bivariate sample. It is important to note that the shape of the Young tableau only depends on the copula of the joint distribution of the two random variables and not on the marginal distributions. Because of this, the probability distribution of the shape of the Young tableau for the case of independence is unique.

[1] Garcia JE, Gonzalez-Lopez VA (2020) Random Permutations, Non Decreasing Subsequences and Statistical Independence. Symmetry, 12, 9, 1415. https://doi.org/10.3390/sym12091415
[2] Garcia JE, Gonzalez-Lopez VA (2014) Independence tests for continuous random variables based on the longest increasing subsequence. Journal of Multivariate Analysis, 127, 126–146. https://doi.org/10.1016/j.jmva.2014.02.010