If I understand correctly, in a logistic regression, we have that $Y_i \mid X \sim Bern(S(X\beta))$ where $S(x)$ is the sigmoid function.

Suppose we estimate $\beta$ using MLE and get $\hat \beta$. Now, from what I found online, $\hat\beta$ doesn't have a closed-form solution and is estimated by numerically solving $\partial \ell/\partial\beta = 0$ where $\ell(X\beta; \mathbf y)$ is the log-likelihood function (is this correct?).

Given that we don't even have a closed-form solution for this estimator, do we know the sampling distribution of $\hat\beta?$

How do we get asymptotic distribution for $\hat\beta$ (somewhere I think CLT will be used but how)?



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