Relative accuracy of two methods for approximating observed Fisher information

The Fisher information matrix (FIM) has long been of interest in statistics and other areas. It is widely used to measure the amount of information and calculate the lower bound for the variance for maximum likelihood estimation (MLE). In practice, we do not always know the actual FIM. This is often because obtaining the firstor second-order derivative of the log-likelihood function is difficult, or simply because the calculation of FIM is too formidable. In such cases, we need to utilize the approximation of FIM. In general, there are two ways to estimate FIM. One is to use the product of gradient and the transpose of itself, and the other is to calculate the Hessian matrix and then take negative sign. Mostly people use the latter method in practice. However, this is not necessarily the optimal way. To find out which of the two methods is better, we need to conduct a theoretical study to compare their efficiency. In this paper, we mainly focus on the case where the unknown parameter that needs to be estimated by MLE is scalar, and the random variables we have are independent. In this scenario, FIM is virtually Fisher information number (FIN). Using the Central Limit Theorem (CLT), we get asymptotic variances for the two methods, by which we compare their accuracy. Taylor expansion assists in estimating the two asymptotic variances. A numerical study is provided as an illustration of the conclusion. The next is a summary of limitations of this paper. We also enumerate several fields of interest for future study in the end of this paper.