Comparative Confidence Regions using Expected and Observed Fisher Information

Maximum likelihood estimation (MLE) is a fundamental method for statistical prediction, with confidence regions playing a critical role in assessing the accuracy of these estimates. Established theory shows that the standardized MLE is asymptotically normally distributed, with its covariance matrix approximated by either the inverse of the observed Fisher information matrix (FIM) or the inverse of the expected FIM. This paper demonstrates that the expected FIM provides a more accurate approximation for the MLE confidence region under typical conditions. Specifically, in asymptotic settings, the eigenvalues and eigenvectors of the expected FIM evaluated at the MLE exhibit lower mean squared error relative to the true covariance matrix compared to those derived from the observed FIM. These conclusions are supported by theoretical analysis and numerical experiments across multiple scenarios.