Does a maximal coupling distribution have the highest correlation?

Question

Sampling from a maximal coupling of 2 univariate Gaussians will give you a bivariate distribution which is not bivariate Gaussian. A bivariate Gaussian can be parameterized with a correlation parameter $\rho$. I assume that the maximal coupling distribution will have a larger correlation than the bivariate Gaussian but I might be wrong on this-the maximal coupling could be less correlated than the bivariate Gaussian. Are there any results on the relationship between maximal coupling and correlation?

Answer 1

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

The correlation associated with a maximal coupling can be derived from \begin{align*}\mathbb{E}[XY]&=\mathbb{E}[XY\mathbb{I}_{X=Y}]+\mathbb{E}[XY\mathbb{I}_{X\ne Y}]\\ &=\int x^2 p(x)\wedge q(x)\text{d}x+ \int\int xy \{p(x)-p(x)\wedge q(x)\}\{q(y)-p(y)\wedge q(y)\}\text{d}x\text{d}y \end{align*}

Note that, to answer the question, the correlation returned by maximal coupling is a function of the four parameters of the Gaussians, while the maximal correlation is always equal to one.