Incorporating support constraints into nonparametric estimators of densities

Let fn ∗ (x) be the usual Parzen-Rosenblatt kernel estimator of the pdf f of a random variable X based on a sample X1,…,Xn from X.In many practical applications,it is knownt hat X>c and/or X<d for given constants c and d.Additionally, one might know the values of(c)and/or f(d).“mirrorimage”and“tieddown”modifications of fn ∗incorporate this additional information into an estimator fn which has support [c,d].This estimatoris interpreted in a manner which allows one to use most of the known convergence properties of kernel estimates in studying the behavior of fn.