Notes and Comments on the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions By

We show by example that empirical likelihood and other commonly used tests for moment restrictions are unable to control the (exponential) rate at which the probability of a Type I error tends to zero unless the possible distributions for the observed data are restricted appropriately. From this, it follows that for the optimality claim for empirical likelihood in Kitamura (2001) to hold, additional assumptions and qualifications are required. Under stronger assumptions than those in Kitamura (2001), we establish the following optimality result: (i) empirical likelihood controls the rate at which the probability of a Type I error tends to zero and (ii) among all procedures for which the probability of a Type I error tends to zero at least as fast, empirical likelihood maximizes the rate at which the probability of a Type II error tends to zero for most alternatives. This result further implies that empirical likelihood maximizes the rate at which the probability of a Type II error tends to zero for all alternatives among a class of tests that satisfy a weaker criterion for their Type I error probabilities.