the approximation of the variance of MLE (Cramer-Rai Lower Bound)

Question

This is in In Casella's Statistical Inference，page 473, the approximation of the variance of MLE (Cramer-Rao Lower Bound). I really confused with the conclusion:

$Var_{\hat{\theta}}h(\hat{\theta})$ is a consistent estimator of $Var_{\theta}h(\hat{\theta}).$

I guess the logic is

$\dfrac{[h'(\theta)]^2}{I_n(\theta)}$ is asymptotic variance of $Var_{\theta}h(\hat{\theta})$ and $Var_{\hat{\theta}}h(\hat{\theta})$ is consistent estimator of $\dfrac{[h'(\theta)]^2}{I_n(\theta)}.$ Then $Var_{\hat{\theta}}h(\hat{\theta})$ is a consistent estimator of $Var_{\theta}h(\hat{\theta}).$

My questions are:

1. Why $-\dfrac{1}{n}\dfrac{\partial^2}{\partial\theta^2}\log L(\theta|\bf{X})\Big|_{\theta=\hat{\theta}}$ is consistent estimator $I(\theta) = \dfrac{1}{n}E_{\theta}[-\dfrac{\partial^2}{\partial\theta^2}\log L(\theta|\bf{X})]?$ since $E_{\theta}[-\dfrac{\partial^2}{\partial\theta^2}\log L(\theta|\bf{X})] \neq -\dfrac{\partial^2}{\partial\theta^2}\log L(\theta|\bf{X}),$ we cannot use the consistency property of MLE.

2. Why A is asymptotic variance of B and C is the consistent estimator of A => C is the consistent estimator of B?

Answer 1

1.$-\dfrac{1}{n}\dfrac{\partial^2}{\partial\theta^2}\log L(\theta|\bf{X})\Big|_{\theta=\hat{\theta}}$ is a consistent estimator of $I(\theta)$ since it converges in probability to the true value $\dfrac{1}{n}E_{\theta}[-\dfrac{\partial^2}{\partial\theta^2}\log L(\theta|\bf{X})]$. What you have pointed out is the condition for an estimator to be unbiased not consistent. Consistency and unbiasdness of estimators are different concepts.

2.could you please elaborate more on this?

Hope this helps