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?
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1$\begingroup$ The question does not make complete sense in that there is no $\rho$ involved in the statement of the maximal coupling, only marginals matter. $\endgroup$ – Xi'an Dec 25 '17 at 10:07
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$\begingroup$ @Xi'an I agree the the marginals are explicitly specified but the procedure mentioned on Pierre's blog also gives an implied joint distribution over $(X,Y)$ does it not? I was thinking for the normal case you might be able to use a conditioning argument to get to a relationship between covariance of the implied distribution over $(X,Y)$ under the 2 cases in Pierre's algorithm. From this you could get a correlation. Incidentally, I was reading Pierre's blog post and came up with this question-I should've included a link in the original post-thanks for including. $\endgroup$ – Lucas Roberts Dec 26 '17 at 16:29
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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.
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1$\begingroup$ Marking as correct, thanks for putting this together. $\endgroup$ – Lucas Roberts Dec 27 '17 at 15:50
