simple condition for Asymptotic normailty of MLE

Question

Is these condition sufficient for asymptotic normality of MLE? if it is pls. let me know the references. 1-First and second derivatives of $\ell(\theta,\eta)$ are defined. 2-The Fisher information matrix be non-singular and continuous with respect to the parameters $\theta$ and $\eta$.

Answer 1

No. You need (among other conditions, but this one is the most crucial): the Hessian of the log-likelihood function $l(\theta, x)$ needs to be dominated by a $L^1$-function

$$ \sup_{\theta \in N} \| \frac{\partial ^2}{ \partial ^2 \theta} l(\theta, x) \| \leq d(x), \; d \in L^1. $$

where $\theta \in \Theta \subset \mathbb{R}^p$ is the parameter (for you $p = 2$), $\frac{\partial ^2}{ \partial \theta ^2}(\cdot)$ means computing the Hessian, $\| \cdot \|$ is any reasonable norm, and $N$ is a neighborhood containing the true parameter.