On the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions
通过反例指出经验似然等矩条件检验无法控制第一类错误概率趋于零的速率，除非对数据分布施加限制；在更强假设下，证明了经验似然能最大化第二类错误概率趋于零的速率，从而修正了Kitamura（2001）的最优性结论。
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.