I'm working on a question and I appreciate if you could guide me on how to approach it. Here is the question:
Consider $Y_1, Y_2, \ldots, Y_n$ as iid with density $f(y;\theta)$ and assume that the first and the second derivative of $log f$ with respect to $\theta$ exists. Derive the asymptotic distribution of $\hat{\theta}$ and suggest a consistent estimator of it's asymptotic variance. Also, consider a g function such that:
- $E_{\theta}[g(Y;\theta)] = 0$
- $\partial{g(Y;\theta)}$ exists,
- $E_\theta[g^2(Y;\theta)] < \infty \text{ } \forall \theta$
My solution: The first step is to write the likelihood function:
$$L(\theta) = \prod_{i = 1}^n f(y_i;\theta) \rightarrow \log (L) = \sum_{i = 1}^n f(y_i;\theta) \rightarrow \\ U(\theta) = \text{score function} = \frac{\partial{\log(L)}}{\partial{\theta}} = \sum_{i = 1}^n f'(yi ; \theta) = 0$$
I don't know how to continue. I don't know whether I can use properties of MLE saying that $\theta $ is normal or I should prove that it's normal.
I appreciate your help.
