Sequential Nonparametric Estimation with Assigned Risk

Abstract

The problem is to estimate sequentially a nonparametric function known to belong to an $\alpha$-th-order Sobolev subspace $(\alpha > \frac{1}{2})$ 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 $\gamma < \alpha \leq 2\gamma$ for some given $\gamma \geq \frac{1}{2}$.