Global consistency of empirical likelihood

Empirical likelihood is a popular nonparametric framework for inference due to its flexibility in model assumptions. The resulting profile empirical likelihood function shares similar properties with the parametric likelihood. According to Qin and Lawless [23], there exists a consistent local maximum within a small neighbourhood of the true parameter value. However, it is usually unknown whether a given local maximum is close to the true parameter value. To fill this important gap, we identify a set of conditions under which the global maximum of the profile empirical likelihood function is consistent. We subsequently propose a ‘global consistency remedy’ to ensure this property. We further propose a ‘global maximum test’ to ascertain whether a given local maximum is, in fact, the global maximum.