Abstract: Empirical likelihood is a general non-parametric inference methodology. It uses likelihood principle in a way that is analogous to that of parametric likelihoods. In a wide range of applications the methodology was shown to provide likelihood ratio statistics that have limiting chi-square distributions and yield a nonparametric version of Wilks theorem. Amongst recent extensions of empirical likelihood are the analysis of censored data, longitudinal data and semi-parametric regression models. Wilks theorem remains true in some, but not in others. This motivates our comparison of extended empirical likelihood methods. We evaluate their relative optimality by comparing the confidence regions provided by inverting the likelihood ratio tests. We show that those extension methods with the likelihood ratio statistic observing the Wilks theorem provide the smallest confidence regions. Specific examples are provided for the case of censored data analysis and estimating equations involving nuisance parameters.

Key words and phrases: Empirical likelihood, likelihood ratio, scaled chi square distribution, Wilks confidence region.