Inference on parameters estimated using optimization algorithm I don't have too much knowledge about mathematical statistics, so I would like to know how can I derive the confidence intervals and the distribution according a parameter(s) that was estimated using some optimization algorithm or numerical approximation with MLE.
For example, in the Spatial Auto-regressive Model I can derive the distribution of the vector beta using the estimator (a closed formula), but the auto-regressive spatial parameter is estimated using a numerical method, but I don't know how to derive the distribution according with the "estimator" in this case.
 A: Typically these sampling distributions are asymptotically multivariate normal, e.g. https://stats200.stanford.edu/Lecture14.pdf:

The goal of this lecture is to explain why ... consistency and asymptotic normality of the MLE hold quite generally for many
  “typical” parametric models, and there is a general formula for its asymptotic variance. The
  following is one statement of such a result:
Theorem 14.1. Let $\{f (x|θ) : θ ∈ Ω\}$ be a parametric model, where $θ ∈ R$ is a single IID
  parameter. Let $X_1 , \ldots , X_n ∼ f (x|θ_0 )$ for $θ_0 ∈ Ω$, and let $\hat θ̂$ be the MLE based on $X_1 , \ldots , X_n$ .
  Suppose certain regularity conditions hold, including: 
  • All PDFs/PMFs $f (x|θ)$ in the model have the same support,
  • $θ_0$ is an interior point (i.e., not on the boundary) of $Ω$,
  • The log-likelihood $l(θ)$ is differentiable in $θ$, and
  • $\hat θ̂$ is the unique value of $θ ∈ Ω$ that solves the equation $0 = l' (θ)$.
  Then $\hat θ̂$ is consistent and asymptotically normal, with 
  $\sqrt{n}(\hat θ- θ_0) \to N\left(0, \frac{1}{I(θ_0)} \right)$ in distribution.

In particular, $I$ here is the information matrix, which you can estimate numerically from the second partial derivatives (e.g. by finite differences) of the log-likelihood function.
