Level-specific correction for nonparametric likelihoods

The popular empirical likelihood method not only has a convenient chi-square limiting distribution but is also Bartlett correctable, leading to a high-order coverage precision of the resulting confidence regions. Meanwhile, it is one of many nonparametric likelihoods in the Cressie-Read power divergence family. The other likelihoods share many attractive properties but are not Bartlett correctable. In this paper, we develop a new technique to achieve the effect of being Bartlett correctable. Our technique is generally applicable to pivotal quantities with chi-square limiting distributions. Numerical experiments and an example reveal that the method is successful for several important nonparametric likelihoods.