I've posted this already in stats.stackexchange. I'm not sure what the rules are for cross-posting but mathoverflow seems to be more active.


Suppose we have data $x\_i, i=1,2,3,...n$ that are *dependent* and identically distributed with marginal $f(\cdot|\alpha)$. If we model this with the likelihood


$
L = c(F(x\_1|\alpha),F(x\_2|\alpha),...F(x\_n|\alpha)|\theta)\prod\_{i=1}^n f(x\_i|\alpha)
$


and the dependence parameter $\theta$ is known, can we apply some variant of the Expectation Maximization algorithm to estimate $\alpha$ using an iterative procedure with relatively simple steps?


For instance, I considered a simple problem with exponential marginals and Gaussian copula (with known correlation), and did something procedural (and hokey). I introduced the unknown independent samples $\tilde{x}\_i, i=1,2,...,n$ which you would compute by knowing the correct value of $\alpha$, mapping the $x\_i$ to correlated Gaussians $y\_i = \Phi^{-1}(F(x\_i|\alpha))$ and then "undoing" the correlations $z = B^{-1}y$ and mapping the $z$ forward again to produce the independent $\tilde{x\_i}=F^{-1}(\Phi(z\_i)|\alpha)$. Here $C=BB^T$ is the correlation matrix. Here $\Phi$ is the (0,1)-normal cdf. If you turn this into an iterative procedure, using $x$ as the initial guess for the independent data, it seems to produce a series that (at least in my trials) converged. However, the whole thing is doubtful since it depends entirely on what you choose for $B$ (only defined up to a unitary matrix). I think it's the unitary invariance of the Gaussian hitting you when you try to basically do an inversion.


Is there an obvious way to turn this kind of problem into a sane iterative procedure using simple steps like the EM? I feel like I'm missing a simple trick.



marginal parameter estimation in copula with copula (dependence) parameter known
The maximization by parts algorithm for maximum likelihood estimation is capable of exactly handling that situation


[Peter X.-K Song, Yanqin Fan & John D Kalbfleisch (2005), Maximization by Parts in Likelihood Inference, Journal of the American Statistical Association.](http://pubs.amstat.org/doi/pdf/10.1198/016214505000000204)


One of the examples in that paper is slightly more general than the copula setting in your question (it does not assume $\theta$ is known, so I would expect the implementation in your case to be simpler than that example).


