On nonparametric regression for IID observations in a general setting

We consider the problem of sharp-optimal estimation of a response function $f(x)$ 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)) π(x). We wish to estimate the response function with optimal minimax mean integrated squared error convergence as the sample size tends to ∞. Traditional regularity assumptions on the conditional density p(y| θ) assumed for parameter θ estimation are sufficient for sharp-optimal nonparametric risk convergence as well as for the existence of the best constant and rate of risk convergence. This 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 sequential approach and some aspects of experimental design are considered as well.