Constructing nonparametric likelihood confidence regions with high order precisions

Abstract

Empirical likelihood is a natural tool for nonparametric statistical infer- ence, and a member of nonparametric likelihoods. Inferences based on this class of likelihoods have the same first order asymptotic properties. One member of the class, exponential tilting likelihood, has been found to be stable to model mis- specification but is not as efficient as empirical likelihood. Exponentially tilted empirical likelihood, also called exponential empirical likelihood, was proposed to achieve both stability and efficiency. Unlike empirical likelihood, however, the hy- brid likelihood is not Bartlett correctable, and the precision of its confidence regions is compromised when the sample size is not large. We introduce a novel adjustment procedure and show that it attains the high order precision that is not attained by the usual Bartlett correction. Simulation results confirm the improved precision in coverage probabilities.