Density estimation under constraints

We suggest a general method for tackling problems of density estimation und er constraints. It is, in effect, a particular form of the weighted bootstr ap, in which resampling weights are chosen so as to minimize distance from the empirical or uniform bootstrap distribution subject to the constraints being satisfied. A number of constraints are treated as examples. They incl ude conditions on moments, quantiles, and entropy, the latter as a device f or imposing qualitative conditions such as those of unimodality or "interes tingness." For example, without altering the data or the amount of smoothin g, we may construct a density estimator that enjoys the same mean, median, and quartiles as the data. Different measures of distance give rise to slig htly different results.