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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)?

Dayne
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    Related: https://stats.stackexchange.com/questions/27322/are-logistic-regression-coefficient-estimates-biased-when-the-predictor-has-larg https://stats.stackexchange.com/questions/303180/standard-error-of-the-estimate-in-logistic-regression https://stats.stackexchange.com/questions/258788/normally-distributed-estimated-parameters-in-logistic-regression https://stats.stackexchange.com/questions/85682/how-does-maximum-likelihood-estimation-have-an-approximate-normal-distribution – kjetil b halvorsen Dec 14 '20 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. – Dayne Dec 14 '20 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? – Dayne Dec 14 '20 at 13:40
  • Your extra question is difficult without links/references--- but you can ask it as a new question, with those links – kjetil b halvorsen Dec 14 '20 at 15:37

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