# Sampling/Asymptotic Distribution of Estimated Coefficients of Logistic Regression

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)?

• Commented Dec 14, 2020 at 12:10
• @kjetilbhalvorsen: thanks a lot for these references. The second link seem to answer my question (will look more into in detail as I have some doubts there). Fourth is also very interesting, as it mentions that MLE for any model follows normal asymptotically. So thanks again. Commented Dec 14, 2020 at 13:38
• Also, I found online that there are some other ways of estimating coefficients in logistic regression. Does any such (accepted) method have a sampling distribution? Commented Dec 14, 2020 at 13:40
• Your extra question is difficult without links/references--- but you can ask it as a new question, with those links Commented Dec 14, 2020 at 15:37
• Does this answer your question? Bias of maximum likelihood estimators for logistic regression Commented Sep 27, 2023 at 0:28