An almost-parametric estimate of regression (Q1112502)

We study the estimation of a function f, defined on an abstract space X, from observations with additive noise \(Y=f(x)+\epsilon\); the points of the plan x are chosen in a special way. We are interested in a scheme in which the function f belongs to a linear manifold of finite, yet unknown dimension, and the noises are independent and not identically distributed. In this ``almost parametric'' variant of nonparametric estimation of regression we construct a \(\sqrt{n}\)-consistent (asymptotically normal if the dispersion of the noise is constant) estimate of the regression function at a point. As an intermediate result we construct a consistent estimate of the dimension of the model.