Adjusted empirical likelihood and its properties

Computing a profile empirical likelihood function, which involves constrained maximization, is a key step in applications of empirical likelihood. However, in some situations, the required numerical problem has no solution. In this case, the convention is to assign a zero value to the profile empirical likelihood. This strategy has at least two limitations. First, it is numerically difficult to determine that there is no solution; second, no information is provided on the relative plausibility of the parameter values where the likelihood is set to zero. In this article, we propose a novel adjustment to the empirical likelihood that retains all the optimality properties, and guarantees a sensible value of the likelihood at any parameter value. Coupled with this adjustment, we introduce an iterative algorithm that is guaranteed to converge. Our simulation indicates that the adjusted empirical likelihood is much faster to compute than the profile empirical likelihood. The confidence regions constructed via the adjusted empirical likelihood are found to have coverage probabilities closer to the nominal levels without employing complex procedures such as Bartlett correction or bootstrap calibration. The method is also shown to be effective in solving several practical problems associated with the empirical likelihood.