This article discusses univariate density estimation in situations when the sample (hard
information) is supplemented by “soft” information about the random phenomenon. These situations
arise broadly in operations research and management science where practical and computational reasons
severely limit the sample size, but problem structure and past experiences could be brought in. In
particular, density estimation is needed for generation of input densities to simulation and stochastic
optimization models, in analysis of simulation output, and when instantiating probability models. We
adopt a constrained maximum likelihood estimator that incorporates any, possibly random, soft information
through an arbitrary collection of constraints. We illustrate the breadth of possibilities by
discussing soft information about shape, support, continuity, smoothness, slope, location of modes,
symmetry, density values, neighborhood of known density, moments, and distribution functions. The
maximization takes place over spaces of extended real-valued semicontinuous functions and therefore
allows us to consider essentially any conceivable density as well as convenient exponential transformations.
The infinite dimensionality of the optimization problem is overcome by approximating splines
tailored to these spaces. To facilitate the treatment of small samples, the construction of these splines
is decoupled from the sample. We discuss existence and uniqueness of the estimator, examine consistency
under increasing hard and soft information, and give rates of convergence. Numerical examples
illustrate the value of soft information, the ability to generate a family of diverse densities, and the
effect of misspecification of soft information.U.S. Army Research Laboratory and the U.S. Army Research Office grant 00101-80683U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-10-1-0246U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-12-1-0273U.S. Army Research Laboratory and the U.S. Army Research Office grant 00101-80683U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-10-1-0246U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-12-1-027
