Minimax Confidence Intervals for Pointwise Nonparametric Regression Estimation

We address a problem of estimation of an unknown regression function f at a given point x0 from noisy observations yi = f(xi)+ ei, ;i =1, ..., n. Here xi in ε Rk are observable regressors and (e_i) are normal i.i.d. (unobservable) disturbances. The problem is analyzed in the minimax framework, namely, we suppose that f belongs to some functional class F, such that its finite-dimensional cut Fn = {f(xi), f ε F, i =0, ..., n, } is a convex compact set. For an arbitrary fixed regression plan Xn =(x1;...;xn) we study minimax on Fn confidence intervals of affine estimators and construct an estimator which attains the minimax performance on the class of arbitrary estimators when the confidence level approaches 1.CADIC