  Maximum likelihood estimates and corresponding confidence regions of the
estimates are commonly used in statistical inference. In practice, people often
construct approximate confidence regions with the Fisher information at given
sample data based on the asymptotic normal distribution of the MLE (maximum
likelihood estimate). Two common Fisher information matrices (FIMs, for
multivariate parameters) are the observed FIM (the Hessian matrix of negative
log-likelihood function) and the expected FIM (the expectation of the observed
FIM). In this article, we prove that under certain conditions and with an MSE
(mean-squared error) criterion, approximate confidence interval of each element
of the MLE with the expected FIM is at least as accurate as that with the
observed FIM.
