1 research outputs found
Maximum Likelihood under constraints: Degeneracies and Random Critical Points
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. when the null hypothesis is not true. We establish
that the behaviour of the optimal distribution under such mis-specification
differ markedly from their properties under the null, i.e. when the estimating
equations are unbiased and correctly specified. This is manifested by certain
degeneracies in the optimal distribution which define the likelihood. Such
degeneracies are not observed under the null. Furthermore, we establish an
anomalous behaviour of the log-likelihood based Wilks statistic, which, unlike
under the null, does not exhibit a chi-squared limit. In the Bayesian setting,
we rigorously establish the posterior consistency of procedures based on these
ideas, where instead of a parametric likelihood, an empirical likelihood is
used to define the posterior distribution. In particular, we show that this
posterior, as a random probability measure, rapidly converges to the delta
measure at the true parameter value. A novel feature of our approach is the
investigation of critical points of random functions in the context of such
empirical likelihood. In particular, we obtain the location and the mass of the
degenerate optimal weights as the leading and sub-leading terms in a canonical
expansion of a particular critical point of a random function that is naturally
associated with the model.Comment: 45 pages, 3 figure