Nonparametric regression estimation with assigned risk

In a controlled random design experiment it is often desirable to estimate an underlying regression function with assigned risk, then the optimality is understood as a minimal mean sample size. For the first time in the literature, it is proved that such a sequential procedure exists. Further, this procedure is adaptive to the unknown design density and scale function. Further, the procedure is robust to the distribution of regression errors. The theoretical results prove several conjectures of Efromovich [Efromovich, S., 2007. Sequential design and estimation in heteroscedastic nonparametric regression. Sequential Anal. 25, 3–26].