On Wikipedia's page for the wrapped Cauchy distribution, there's a fixed-point algorithm to calculate the maximum-likelihood estimator using a Möbius transformation in the Poincaré disc that the circular Cauchy distribution can be considered to lie in.

By using the parameterization $\theta=x_0+i\gamma$ on the regular Cauchy distribution and treating $\theta,\overline{\theta}$ as independent variables, the maximum likelihood equation becomes $$\sum_{j=1}^n\frac{1}{\theta-x_j}=\frac{n}{\theta-\overline{\theta}} $$ and there's a similar (likely the same after Cayley transform, even) estimator $$\theta_{k+1}=\overline{\theta_k-n\left(\sum_{j=1}^n \frac{1}{\theta_k-x_j} \right)^{-1}} $$ which can also be shown to be a contraction in the Poincaré upper half-plane metric by Schwarz-Pick (and that $z \mapsto -\overline{z}$ is an isometry). However, the Wikipedia page's sources do not mention these and I've been unable to find a reference showing either algorithm, aside from Wikipedia itself. But, the calculation is simple enough that I'd be legitimately surprised if neither had been done before. Since I'm not hugely familiar with the statistics literature, I figure this should be aired out.



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