ON NONPARAMETRIC REGRESSION FOR IID OBSERVATIONS IN A GENERAL SETTING

we consider the problem of sharp-optimal estimation of a response func tion f(x) in a random design nonparametric regression under a general model where a pair of observations (Y, X) has a joint density p(y, x) = p(y\f(x))pi(x). We wish to estimate the response function with optim al minimax mean integrated squared error convergence as the sample siz e tends to infinity. Traditional regularity assumptions on the conditi onal density p(y\theta) assumed for parameter theta estimation are suf ficient for sharp-optimal nonparametric risk convergence al well as fo r the existence of the best constant and rate of risk convergence. Thi s best constant is a nonparametric analog of Fisher information. Many examples are sketched including location and scale families, censored data, mixture models and some well-known applied examples. A sequentia l approach and some aspects of experimental design are considered as w ell.