Nonparametric maximum likelihood estimation of a probability density via mathematical programming

Based on sample values of a one-dimensional random variable a nonparametric maximum likelihood estimate for the unknown probability density is introduced as the solution of an optimization problem in an appropriate Hilbert space. This solution turns out to be a polynomial spline function, and a complete characterization is given using recent results on the differentiability of the optimal value of a parametrized family of optimization problems. An important feature of this estimate is that its support interval results in a quite natural way from the formulation of the problem and is not fixed in advance. The estimator is shown to have a certain consistency property for a special class of density functions. Numerical results will be given in a subsequent paper.