Comparison between Expected and Observed Fisher Information in Interval Estimation

Maximum likelihood estimates and corresponding confidence regions of the estimates are commonly used in statistical inference. In practice, people usually construct approximate confidence regions with the Fisher information at given sample data based on the asymptotic normal distribution of the MLE (maximum likelihood estimator). 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 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.