# When is logistic regression MLE consistent and asymptotically normal?

Logistic regression model is defined as

$$\mathbb{P}\left\{ Y = 1\mid X\right\}=\frac{e^{\beta^T X}}{1 + e^{\beta^T X}}$$

where $Y$ is a random variable in $\left\{0,1\right\}$, and $X$ is a random vector in $\mathbb{R}^d$. Consider the MLE of this model. Under what conditions is the MLE estimator $\hat{\beta}_n$ consistent (i.e. $\hat{\beta}_n \overset{p}\to \beta_0$), and $\sqrt{n}\left(\hat{\beta}_n - \beta_0\right)$ is asymptotically normally distributed?

The theorem I am looking at (which is provided in the lecture but without proof) is

Following this theorem, the conditions I came up with were

1. Identifiability
2. $\mathbb{E}\left[ \left| X_i\right|^2\right]$ is finite for $i=1,\cdots,d$.
3. The matrix $\mathbb{E}\left[XX^T\right]$ is non-singular.

I am wondering if there is any reference on the conditions. I don't know if I'm right or not, partly because I can't find this theorem somewhere else.

• @Xi'an I have read them, but seems like none of them match the theorem I am looking at. For example, I don't understand why the first reference requires something on eigenvalue. – 3x89g2 Feb 27 '18 at 15:00
• @Xi'an I have also read the paper by Shifa, but I don't understand why they use a theorem with so many assumptions, instead of the one we covered in the lecture? – 3x89g2 Feb 27 '18 at 16:23
• What about asking the lecturer if it comes from a lecture! – Xi'an Feb 27 '18 at 17:45
• @Xi'an Good point... She seems reluctant to answer questions, but I guess I will try again... – 3x89g2 Feb 27 '18 at 17:57
• You may want to check out Theorem 4.2.1 and 4.2.2 of Herman Bierens' book "Topics in advanced econometrics". They are standard M estimator results. – semibruin Mar 7 '18 at 11:41