log of maximum likelihood function.

Question

Given 100 observations and density function : $ f(x;\theta) = \frac{1}{\pi \cdot \left(1+\left(x-\theta \right)^2\right)}$

find $\hat{\theta}_{MLE}$.

My Attempt:

I have found the joint likelihood $L(\theta;x_1,x_2\dots x_{100}) = \prod _{i=1}^{100}\left(\frac{1}{\pi \cdot \left(1+\left(x_i-\theta \right)^2\right)}\right)\:$

I have tried to find $l=\log(L)$ but I failed.

Answer 1

There's no good form of an answer. In cases $n=1$ and $n=2$ there's a simple way to write an answer, but in case of big $n$ we may write only $$\theta = argmax_{\theta \in \Theta} L(\theta, x_1, \ldots, x_{n}).$$ The problem is that we can not find the minimum of polynomial of degree $2n= 200$ explicitly.