An evaluation of the power and conditionality properties of empirical likelihood

Empirical and dual likelihood are two of a growing array of artificial or approximate likelihoods currently in use in statistics. A question of major interest focuses on the performance of these new constructs relative to ordinary parametric likelihoods. We consider here two criteria, local power and conditional properties. Looking at tests of a scalar parameter, we show that there is no loss of efficiency in using a dual or empirical likelihood model, to second order. To third order, either the artificial likelihood or the true likelihood test could be more efficient; this is determined by the Fisher information, the distance between the null and the alternative hypotheses, and the statistical curvature of the models. Conditionality properties are assessed by comparing empirical likelihood with quasilikelihood. If there is overdispersion present, or more generally an unknown amount of dispersion, then there is little difference in the ancillaries for conditional inference based on empirical likelihood or on other possible likelihoods, such as a quasilikelihood. If the amount of dispersion is known, however, it is correct to base inference on the quasilikelihood.