A Relationship between Classical and Bayesian Estimation Procedures through Fisher Information

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The aim of this article is to investigate the association between the classical and Bayesian approaches through Fisher information. For any particular distribution, the computation of Fisher information is quite significant, as it provides the amount of information about the unknown parameter inferred from the observed data and is related to classical methods of estimation. Also, in the light of some prior knowledge, we may estimate the unknown parameter through Bayesian approach. Specifically, we want to see a relationship between information and Bayes estimation. In this article, the scale parameter of the one-parameter exponential distribution is estimated under the weighted squared error and Kullback-Leibler distance loss functions. The information acquired from both the classical and Bayesian methodologies have been connected through the risk intensity and error intensity which have been introduced in this article. The results of extensive simulation studies using these intensity measures show that the Bayes estimator performs more intensely as the amount of Fisher information increases. It is seen that the Fisher information, which is pivotal to many classical estimation methods, has a relationship with the Bayesian method depending on prior distribution, at least in this case, as the intensity measures of the Bayes estimator decrease with the increase in information. Further, to comprehend the theoretical notion of association, two real-life datasets have been included to show usefulness in practical field.