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I think this is a rather open question. Suppose I have bi-dimensional data $(x_i, y_i)$. I have some reasonable model for the marginals, say distributions $F_X$ and $F_Y$ (parametric).

How to reasonably construct a bi-dimensional model for the joint distribution of $(X,Y)$ that respects say, the correlation between $X$ and $Y$ or some other higher moment? Of course I'm not considering normal variates.

kjetil b halvorsen
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    Are you familiar with [tag:copula]s? – whuber Jul 18 '13 at 21:04
  • The bidimensional data that you have _are_ a model for the joint distribution of $X$ and $Y$, and the corresponding sample marginal distributions are a model for the marginal distributions of $X$ and $Y$. Incidentally, you say that you are not considering normal variates but even if you are, estimating a joint normal distribution from the sample correlation is also model-building where you have rejected all other joint distributions in favor of the joint normal, even though, as [cardinal points out](http://stats.stackexchange.com/a/30205/6633), this is the exception and not the rule. – Dilip Sarwate Jul 18 '13 at 21:14
  • Some people would approach this as finding the max-ent joint distribution with the given marginals and the given correlation (and maybe some other stats). See https://arxiv.org/pdf/1212.0440.pdf and http://yaroslavvb.com/papers/abbas-entropy.pdf – kjetil b halvorsen May 07 '17 at 10:42

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