Maximum Empirical Likelihood: Empty Set Problem

In the Empirical Estimating Equations (E^3) approach to estimation and inference estimating equations are replaced by their data-dependent empirical counterparts. It is odd but with E^3 there are models where the E^3-based estimator does not exist for some data set, and does exist for others. This depends on whether or not a set of data-supported probability mass functions that satisfy the empirical estimating equations is empty for the data set. In a finite sample context, this unnoted feature invalidates methods of estimation and inference, such as the Maximum Empirical Likelihood, that operate within E^3. The empty set problem of E^3 is illustrated by several examples and possible remedies are discussed.