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.
