Confidence Intervals with Expected and Observed Fisher Information in the Scalar Case

Maximum likelihood estimations (MLEs) and corresponding confidence intervals are commonly used in statistical inference. In practice, people usually construct approximate confidence intervals with the Fisher information at given sample data based on the asymptotic normal distribution of MLE. Two common Fisher information numbers (FINs, for scalar parameters) are the observed FIN (the second derivative of negative log-likelihood function) and the expected FIN (the expectation of the observed FIN). In this article, we prove that under certain conditions and with MSE criterion, approximate confidence intervals with the expected FIN are more accurate than those with the observed FIN. The fact is illustrated in a numerical study related to a standard signal-plus-noise problem.