I am trying to find the MLE of a shifted logistics distribution with the following pdf:
$$f(x;\theta)=\frac{e^{-(x-\theta)}}{(1+e^{-(x-\theta)})^2}$$
Since all my observations are IID the likelihood function is:
$$L(\theta|x)=\prod_{i=1}^{n}\frac{e^{-(x_i-\theta)}}{(1+e^{-(x_i-\theta)})^2}$$
Which I can't find a closed form for the solution. I was asked to prove the uniqueness of the solution of the likelihood equation. Where the likelihood equation is:
$$L(\theta)=\frac{\partial}{\partial \theta}ln\bigg(\prod_{i=1}^{n}\frac{e^{-(x_i-\theta)}}{(1+e^{-(x_i-\theta)})^2}\bigg)=0$$
I think I can show uniqueness without finding the MLE, but the hw question also asks to find the asymptotic distribution of
$$\sqrt{n}(\hat{\theta}-\theta)$$
Where $\hat{\theta}$ is the MLE estimate. I believe I need the mle in order to figure out the asymptotic variance. Is there a way to solve this problem without solving for the MLE? Is there a closed form to the MLE or can it only be found numerically?