The empirical likelihood prior applied to bias reduction of general estimating equations

TL;DR

This paper introduces an empirical likelihood prior that asymptotically maximizes mutual information, enabling bias reduction in solutions of general estimating equations and M-estimators, with practical application demonstrated in a myocardial infarction study.

Contribution

It derives a Jeffreys-type empirical likelihood prior and develops a methodology for asymptotic bias reduction in estimating equations and M-estimation schemes.

Findings

The EL prior asymptotically maximizes Shannon mutual information.

The method effectively reduces asymptotic bias in estimating equations.

Application to real data shows practical benefits of the approach.

Abstract

The practice of employing empirical likelihood (EL) components in place of parametric likelihood functions in the construction of Bayesian-type procedures has been well-addressed in the modern statistical literature. We rigorously derive the EL prior, a Jeffreys-type prior, which asymptotically maximizes the Shannon mutual information between data and the parameters of interest. The focus of our approach is on an integrated Kullback-Leibler distance between the EL-based posterior and prior density functions. The EL prior density is the density function for which the corresponding posterior form is asymptotically negligibly different from the EL. We show that the proposed result can be used to develop a methodology for reducing the asymptotic bias of solutions of general estimating equations and M-estimation schemes by removing the first-order term. This technique is developed in a similar manner to methods employed to reduce the asymptotic bias of maximum likelihood estimates via penalizing the underlying parametric likelihoods by their Jeffreys invariant priors. A real data example related to a study of myocardial infarction illustrates the attractiveness of the proposed technique in practical aspects. Keywords: Asymptotic bias, Biased estimating equations, Empirical likelihood, Expected Kullback-Leibler distance, Penalized likelihood, Reference prior.