I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's?
Specifically, I am working with the Pareto distribution, which has density function
$$f_X(x)={\alpha k^\alpha \over x^{\alpha +1}},\mbox{ } \alpha,k>0 \mbox{ and } x>k.$$
After computing the MLE estimators $\hat{\alpha}$ of $\alpha$ and $\hat{k}$ of $k$, I was asked to compute the fisher information matrix of $\hat{\alpha}$ and $\hat{k}$.
Best Answer
The Fisher information is essentially the negative of the expectation of the Hessian matrix, i.e. the matrix of second derivatives, of the log-likelihood. In particular, you have $$l(\alpha,k)=\log\alpha + \alpha \log k - (\alpha + 1) \log x$$ from which you compute the second-order derivatives to create a $2 \times 2$ matrix, which you take the expectation (and set $\alpha=\hat \alpha$ and $k=\hat k$).