Probability Density Estimation via Optimal Control

We employ optimal control theory to study the problem of estimating the probability density function from a data set originating from an unknown probability distribution. The original variational problem is reformulated as a multi-stage optimal control problem and the associated maximum principle, or conditions of optimality, is reduced to a two-point boundary-value problem with interior conditions. A numerical scheme is proposed to solve the discretization of this problem. Estimates of density functions for synthetic and real data are computed using the proposed approach. The real data come from the Old Faithful geyser and the speeds of a group of galaxies. Comparisons are made with the popular statistics software R.