It is proved that, except for a location term, empirical likelihood does draw contours which are second-order correct for those of a pseudo-likelihood. However, except in the case of one dimension, this pseudo-likelihood is not that which would commonly be employed when constructing a likelihood-based confidence region. It is shown that empirical likelihood regions may be adjusted for location so as to render them second-order correct. Furthermore, it is proved that location-adjusted empirical likelihood regions are Bartlett-correctable, in the sense that a simple empirical scale correction applied to location-adjusted empirical likelihood reduces coverage error by an order of magnitude. However, the location adjustment alters the form of the Bartlett correction. It is also shown that empirical likelihood regions and bootstrap likelihood regions differ to second order, although both are based on statistics whose centered distributions agree to second order.
"Pseudo-Likelihood Theory for Empirical Likelihood." Ann. Statist. 18 (1) 121 - 140, March, 1990. https://doi.org/10.1214/aos/1176347495