I have two distributions $F$ and $G$ that I conjecture to be identical, mathematically. Essentially, before investing time in mathematically proving that $F$ and $G$ are equal, I'd like to sanity check it by doing a simulation experiment.
So, given that I can easily simulate $X_i \sim F$ and $Y_i \sim G$, I'd like some test that checks $F = G$. The $X_i$ and $Y_i$ live in $\mathbb R^p$, and I'm in the process of checking that the marginals are equal in distribution by simulating many data-sets and checking that the Kolmogov-Smirnov test statistics are approximately uniformly distributed, but I'd like some assurance that I'm not being deceived by just looking at the marginals.
In addition to the marginals, I'm checking the distribution of the principal components of the generated data/some nonlinear transformations of the data, and so far everything points to equality.
If someone could point me to a method - hopefully with an existing implementation in R, or otherwise something easy to implement - that would be fantastic.
EDIT: To be clear, I'm asking for a multivariate test. I know the marginals aren't sufficient, I'm only checking them because if the marginals don't match then I don't need to check the joint.