I'm not at all a statistician, but in the course of my work I've come across a non-trivial maximum likelihood estimation problem, and I'm looking for ideas and/or references on how to approach it. The problem is this:
Points $(x,y)$ are drawn randomly according to $$\begin{eqnarray*} x & = & c_x + \sin(\theta + \phi)\\ y & = & c_y + \sin(\theta) \end{eqnarray*} $$ where $c_x$ and $c_y$ are normally distributed with mean zero and unknown variances, $\theta$ is uniformly distributed on $[0,2\pi]$, and $\phi$ is fixed but unknown. Given many such points $(x,y)$, how can I estimate $\phi$ and the variances of $c_x$ and $c_y$?
Qualitatively, this model places points randomly around an ellipse, with ellipticity determined by $\phi$. The random variables $c_{x,y}$ serve to thicken the ellipse. Actually, this model is a slight simplification of what I'm really working with--in the real model both the amplitude of the $\sin$ terms and $\phi$ also have random components--but I reckon if I can do the above problem then that's good progress.
I've taken a crack at doing the integrals for a direct MLE of the parameters, but it's quite difficult, even numerically. So I figured I should consult experts: Is there an efficient solution to this problem? E.g. perhaps there is some sort of Monte Carlo method for doing this kind of problem?
Edit: It has been suggested that this problem is the same as ellipse fitting. This is incorrect. As explained in this paper (which is where this question is derived from), ellipse fitting gives a biased estimate of $\phi$ and the variances in the above problem. The paper explicitly states that the above MLE problem gives a superior estimate than ellipse fitting. Ellipse fitting is not what this question is asking for.