My problem:

Let $X_1,\dots,X_n \overset{\mathrm{i.i.d}}{\sim} \mathrm{Cauchy}(\theta,1)$ and suppose we want to estimate the location parameter $\theta$. Find the log-likelihood function of the given random sample and show that the maximum likelihood (ML) estimator is solution of an equation of degree $2n - 1$ in $\theta$.

I already have to log-likelihood and $S(\theta)$: enter image description here

But how to show the $2n-1$ thing?

And then the exercise goes on:

Why would this be a problem if we were to find a closed formula for $\hat{\theta}_{ML}$?
What can you say about the uniqueness of such ML estimator in this specific case?

I guess we have no unique solution, right? What else to say? Any hints?

  • 2
    $\begingroup$ When you add the $n$ fractions involved in $S(\theta)$ (which evidently is the derivative of $l(\theta)$ w.r.t. $\theta$, which is the log likelihood for an iid sample of size $n$ from a Cauchy distribution), you obtain a fraction whose numerator is a polynomial of degree $2n-1$ in $\theta$: that's all there is to this question. $\endgroup$ – whuber Oct 7 '13 at 14:27

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