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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.

Yly
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    Consider that for a given $\phi$, the optimal $\theta$ are those that result in points on the ellipse that are closest to the corresponding $x,y$ in terms of some distance function (e.g. Gaussian loss with diagonal variances). So the problem is to find values of $\theta$/variances that make the ellipse "close" in some sense to the data. This is actually a well studied problem in computer vision, referred to as "ellipse fitting" or "elliptical regression". It is usually framed in terms of least squares, see for example http://cseweb.ucsd.edu/~mdailey/Face-Coord/ellipse-specific-fitting.pdf ... – Nate Pope Jul 12 '19 at 20:46
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    @kjetilbhalvorsen I believe this question has been marked a duplicate in error. The linked question concerns ellipse fitting, and I have edited the question body to explain why this is a distinct problem. As explained in the referenced paper, the whole point of setting up the problem as a MLE is to do better than ellipse fitting. – Yly Mar 11 '21 at 23:11
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    Some parts of this edit look confusing: in what sense is $\theta$ "estimated," given it is random? Did you mean $\phi$ is estimated? Do you observe $\theta$ or not? Given that $\phi$ parameterizes a family of ellipses and you are estimating $\phi,$ in what sense is this *not* ellipse fitting? – whuber Mar 11 '21 at 23:24
  • @whuber $\theta$ was a typo, I meant $\phi$; thanks for catching that. My understanding is that ellipse fitting parametrizes the ellipse a different way, as in this post: https://stats.stackexchange.com/questions/268974/ellipse-formula-from-points This results in a biased estimate of $\phi$, which is the quantity of interest here. – Yly Mar 12 '21 at 03:59
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    Right: I appreciate that the different parameterization results in a different probability distribution of the points (which is why I reopened this question). I just wanted to make sure that was your meaning. – whuber Mar 12 '21 at 13:06

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