What is the asymptotic covariance matrix?

Question

Is it true that the asymptotic covariance matrix is equal to the covariance matrix of parameter estimates? If not, what is it? And what is the difference between the covariance matrix and the asymptotic covariance matrix in that case? Thanks in advance!

Answer 1

Given an iid sample $(X_1,\ldots,X_N)$ from a parametric distribution with density $f_\theta(\cdot)$, $\theta$ being the unknown parameter, an estimator $\hat{\theta}(X_1,\ldots,X_N)$ has a distribution with mean $\mu_n(\theta)$ and variance-covariance matrix $\Sigma_n(\theta)$. So $\Sigma_n(\theta)$ is the variance-covariance matrix of $\hat{\theta}(X_1,\ldots,X_N)$ in the sense that $$\mathbb{E}_\theta\left[ \left\{\hat{\theta}(X_1,\ldots,X_N)-\mu_n(\theta)\right\} \left\{\hat{\theta}(X_1,\ldots,X_N)-\mu_n(\theta)\right\}^{\text{T}} \right] = \Sigma_n(\theta)\,.$$

Now, if $\hat{\theta}(X_1,\ldots,X_N)$ is a convergent estimator and if there exists a limiting distribution for $\hat{\theta}(X_1,\ldots,X_N)$, it means there exists a sequence $(\phi_n)$ increasing to $+\infty$, e.g., $\phi_n=\sqrt{n}$, such that $$\phi_n\left\{\hat{\theta}(X_1,\ldots,X_N)-\mu_n(\theta)\right\}\stackrel{\text{dist}}{\longrightarrow} G_\theta$$ where $G_\theta$ denotes a distribution indexed by $\theta$ and the limiting distribution of the l.h.s. This limiting distribution has a variance $\Xi_\theta$ that is called the asymptotic variance.