Asymptotics of MLE without closed form solutions Suppose $L(\theta;X)$ denotes the likelihood of a model where $\theta$ is the parameter and $X$ is the data. $L(\theta;X)$ doesn't have a closed-form solution for MLE. I use a numerical procedure to obtain an estimate $\hat{\theta}$. Then how can I obtain the asymptotics of $\hat{\theta}$?
I am unaware of this area so any reference pointers would also be appreciated.

You can use the fact the MLE is asymptotically unbiased, efficient (i.e. its variance converges to the inverse of the Fisher information), and Gaussian.
In summary, $\hat\theta \rightarrow \mathcal{N}(\theta,\mathcal{I}^{-1}(\theta))$ as the sample size, $n$, goes to infinity.
You can then approximate $\mathcal{I}(\theta)$ by $\mathcal{I}(\hat\theta)$ (the information evaluated at the MLE) to construct a confidence interval for $\theta$, etc.
