Maximum likelihood estimator of Cauchy distribution but with a catch I have an exercise to solve that states that we need to find the Maximum likelihood estimator of location parameter of the Cauchy distribution given a set X={x1,x2} , |x1-x2|<2
Now I worked using this approach: Maximum likelihood estimator of location parameter of Cauchy distribution
but I can't get to grasp how " given a set X={x1,x2} , |x1-x2|<2" changes the problem.
Thank you

To come up with an expression for the conditional likelihood, you express the joint likelihood $L(x_1, x_2)$ as $L(x_2|x_1)L(x_1)$. The unconstrained $x_1$ can be anything of course, so has the usual likelihood
$$f(x_1) = \frac{1}{\pi (1+(x_1-\mu)^2)}$$.
The conditional $x_2$ density can be expressed by assigning likelihood 0 to any value more than 2 units beyond $x_1$, then the density can be "normalized" by the density area within the 2 units, i.e.
$$ f(x_2|x_1) =  \dfrac{f(x_2)}{F(x_1+2) - F(x_1 - 2)} \times \mathcal{I}(|x_1-x_2|<2)$$
I am not sure if the expression $L(x_1, x_2) = f(x_2|x_1) f(x_1)$ has an analytic solution or if it's really necessary to obtain one. But numeric solvers should give you a solution here, and some simulations should show you if the estimator has any nice asymptotic properties.
