Sequential nonparametric estimation of controlled multivariate regression

The article considers an adaptive sequential nonparametric estimation of a multivariate regression with assigned mean integrated squared error (MISE) and minimax mean stopping time when the estimator matches performance of an oracle knowing all nuisance parameters and functions. It is known that the problem has no solution if regression belongs to a Sobolev class of differentiable functions. What if an underlying regression is smoother, say, analytic? It is shown that in this case it is possible to match performance of the oracle. Furthermore, similar to the classical Stein solution for a parameter estimation, a two-stage sequential procedure solves the problem. The proposed regression estimator for the first stage, based on a sample with fixed sample size, is of interest on its own, and a thought-provoking environmental example of reducing potent greenhouse gas emission by an anaerobic digestion system is used to discuss a number of important topics for small samples.