## The Annals of Statistics

- Ann. Statist.
- Volume 23, Number 4 (1995), 1376-1392.

### 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 $\gamma \geq \frac{1}{2}$.

#### Article information

**Source**

Ann. Statist., Volume 23, Number 4 (1995), 1376-1392.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324713

**Digital Object Identifier**

doi:10.1214/aos/1176324713

**Mathematical Reviews number (MathSciNet)**

MR1353510

**Zentralblatt MATH identifier**

0838.62070

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62G07: Density estimation 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators 62J02: General nonlinear regression

**Keywords**

Sequential estimation minimax stopping time curve fitting

#### Citation

Efromovich, Sam. Sequential Nonparametric Estimation with Assigned Risk. Ann. Statist. 23 (1995), no. 4, 1376--1392. doi:10.1214/aos/1176324713. https://projecteuclid.org/euclid.aos/1176324713