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 that in conjunction with a maximum likelihood criterion lead to constrained infinite-dimensional optimization problems that we formulate 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. 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 shape constraints.