Empirical likelihood ratio in penalty form and the convex hull problem

The empirical likelihood ratio is not defined when the null vector does not belong to the convex hull of the estimating functions computed on the data. This may happen with non-negligible probability when the number of observations is small and the dimension of the estimating functions is large: it is called the convex hull problem. Several modifications have been proposed to overcome such drawback: the penalized, the adjusted, the balanced and the extended empirical likelihoods, though they yield different values from the ordinary empirical likelihood in all cases. The convex hull problem is addressed here in the framework of nonlinear optimization, proposing to follow the penalty method. A generalized empirical likelihood in penalty form is considered, and all the proposed modifications are shown to be in that form. We propose simple penalty forms of the empirical likelihood, whose values are equal to those of the ordinary empirical likelihood when the convex hull condition is satisfied. A comparison by simulation and real data is included. All proofs and additional simulation results appear in the Supplementary Material.