Maximum Likelihood under constraints: Degeneracies and Random Critical Points

Understand

We investigate the problem of semi-parametric maximum likelihood under constraints on summary statistics.

• Such a procedure results in a discrete probability distribution that maximises the likelihood among all such distributions under the specified constraints (called estimating equations), and is an approximation to the underlying population distribution.
• The study of such empirical likelihood originates from the seminal work of Owen.
• We investigate this procedure in the setting of mis-specified (or biased) estimating equations, i.e.