Resampling calibrated adjusted empirical likelihood

Empirical-likelihood-based inference for parameters defined by the general estimating equations of Qin & Lawless (1994) remains an active research topic. When the sample size is small and/or the dimension of the accompanying estimating equations is high, the resulting confidence regions often have a lower than nominal coverage probability. In addition, the empirical likelihood can be hindered by an empty set problem. The adjusted empirical likelihood (AEL) tackles both problems simultaneously. However, the AEL confidence region with high-order precision relies on accurate estimation of the required level of adjustment. This has proved difficult, particularly in over-identified cases. In this article, we show that the general AEL is Bartlett-correctable and propose a two-stage procedure for constructing accurate confidence regions. A naive AEL is first employed to address the empty set problem, and it is then Bartlett-corrected through a resampling procedure. The finite-sample performance of the proposed method is illustrated by simulations and an example.