On Maximum Empirical Likelihood Estimation and Related Topics

This article studies maximum empirical likelihood estimation in the case of constraint functions that may be discontinuous and/or depend on additional parameters. The later is the case in applications to semiparametric models where the constraint functions may depend on the nuisance parameter. Our results are thus formulated for empirical likelihoods based on estimated constraint functions that may also be irregular. The key to our analysis is a uniform local asymptotic normality condition for the local empirical likelihood ratio. This condition holds under mild assumptions on the estimated constraint functions and allows for a study of maximum empirical likelihood estimation and empirical likelihood ratio testing similar to that for parametric models with the uniform local asymptotic normality condition. Applications of our results are discussed to inference problems about quantiles under possibly additional information on the underlying distribution, to residual-based inference about quantiles, and to partial adaption.