Maximum Likelihood with Estimating Equations

Methods, like Maximum Empirical Likelihood (MEL), that operate within the Empirical Estimating Equations (E3) approach to estimation and inference are challenged by the Empty Set Problem (ESP). We propose to return from E3 back to the Estimating Equations, and to use the Maximum Likelihood method. In the discrete case the Maximum Likelihood with Estimating Equations (MLEE) method avoids ESP. In the continuous case, how to make ML-EE operational is an open question. Instead of it, we propose a Patched Empirical Likelihood, and demonstrate that it avoids ESP. The methods enjoy, in general, the same asymptotic properties as MEL.