Second-order Accurate Confidence Regions Based on Members of the Generalized Power Divergence Family

Recently, a technique based on pseudo-observations has been proposed to tackle the so-called convex hull problem for the empirical likelihood statistic. The resulting adjusted empirical likelihood also achieves the high-order precision of the Bartlett correction. Nevertheless, the technique induces an upper bound on the resulting statistic that may lead, in certain circumstances, to worthless confidence regions equal to the whole parameter space. In this paper, we show that suitable pseudo-observations can be deployed to make each element of the generalized power divergence family Bartlett-correctable and released from the convex hull problem. Our approach is conceived to achieve this goal by means of two distinct sets of pseudo-observations with different tasks. An important effect of our formulation is to provide a solution that permits to overcome the problem of the upper bound. The proposal, which effectiveness is confirmed by simulation results, gives back attractiveness to a broad class of statistics that potentially contains good alternatives to the empirical likelihood.