Density Estimation under Constraints

Abstract

We suggest a general method for tackling problems of density estimation under constraints. It is, in effect, a particular form of the weighted bootstrap, 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 include conditions on moments, quantiles, and entropy, the latter as a device for imposing qualitative conditions such as those of unimodality or “interestingness.” For example, without altering the data or the amount of smoothing, we may construct a density estimator that enjoys the same mean, median, and quartiles as the data. Different measures of distance·give rise to slightly different results.

Key Words:

• Biased bootstrap
• Cressie—Read distance
• Curve estimation
• Empirical likelihood
• Entropy
• Kernel methods
• Mode
• Smoothing
• Weighted bootstrap