Comparison of Accuracy for Methods to Approximate Fisher Information in the Scalar Case

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 in a set of data and to calculate the lower bound (Cramér−Rao bound) of the variance for estimates such as maximum likelihood and to conduct score tests. (It is also of interest to note that other measures of information in the data are useful in practice, including the Kullback Liebler-Lindley measure, which is derived from the information theoretic notions of Shannon information and the distance between two probability distributions.) In practice, we do not always know the actual FIM. This is often because obtaining the first or second-order derivatives of the log-likelihood function is difficult, or simply because the calculation of FIM is too formidable. In such cases, we need to use 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 better 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 is scalar and the random variables we have are independent. In this scenario, Fisher information matrix 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.