Constrained Maximum Likelihood Estimators for Densities

We put forward a framework for nonparametric density estimation in situations where the sample is supplemented by information and assumptions about shape, support, continuity, slope, location of modes, density values, etc. These supplements are incorporated as constraints and, in conjunction with a maximum likelihood criterion, lead to estimators that are solutions of infinite-dimensional optimization problems formulated over spaces of semicontinuous functions. These spaces, when equipped with an appropriate metric, offer a series of advantages including simple conditions for existence of estimators and their limits and, in particular, guarantee the convergence of modes of densities. Thus, various plug-in estimators emerge naturally from our framework. Relying on the approximation theory---epi-convergence---for optimization problems, we provide general conditions under which estimators subject to essentially arbitrary constraints are consistent and illustrate the framework with a number of examples that span classical and novel constraints.