Empty Set Problem of Maximum Empirical Likelihood Methods

Abstract

In an influential work, Qin and Lawless (1994) proposed a general estimating equations (GEE) formulation for maximum empirical likelihood (MEL) estimation and inference. The formulation replaces a model specified by GEE with a set of data-supported probability mass functions that satisfy empirical estimating equations (E3). In this paper we use several examples from the literature to demonstrate that the set may be empty for some E3 models and finite data samples. As a result, MEL does not exist for such models and data sets. If MEL and other E3-based methods are to be used, then models will have to be checked on case-by-case basis for the absence or presence of the empty set problem.

Keywords: empirical estimating equations, generalized minimum contrast, empirical likelihood, euclidean empirical likelihood, generalized empirical likelihood, affine empty set problem, empirical likelihood bootstrap, model selection