NONPARAMETRIC ESTIMATION OF A SMOOTH DENSITY WITH SHAPE RESTRICTIONS

摘要

Given assumptions about shape and smoothness, a density is estimated non-parametrically using regression splines. Examples of shapes include decreasing, decreasing and convex, and unimodal with known mode. A least-squares criterion is used, so that the estimate is obtained with a single projection onto a convex cone. The convergence rates for the estimators are derived. For the case of unknown mode, a plug-in estimator may be used. If the mode estimator converges fast enough, the rate of the plug-in estimator is the same as for the known-mode estimator. Simulations show that, for small samples, the proposed estimators compare well with competing estimators.