SEQUENTIAL NONPARAMETRIC-ESTIMATION WITH ASSIGNED RISK

The problem is to estimate sequentially a nonparametric function known to belong to an alpha-th-order Sobolev subspace (alpha > 1/2) with a minimax mean stopping time subject to an assigned maximum mean integrated squared error. For the case of a given alpha there exists a sharp estimator which has a minimal constant and a rate of minimax mean stopping time increasing as the assigned risk decreases. The situation changes drastically if alpha is unknown: a necessary and sufficient condition for sharp estimation is that gamma < alpha less than or equal to 2 gamma for some given gamma greater than or equal to 1/2.