## The Annals of Statistics

- Ann. Statist.
- Volume 7, Number 6 (1979), 1196-1221.

### Adaptive Design and Stochastic Approximation

#### Abstract

When $y = M(x) + \varepsilon$, where $M$ may be nonlinear, adaptive stochastic approximation schemes for the choice of the levels $x_1, x_2, \cdots$ at which $y_1, y_2, \cdots$ are observed lead to asymptotically efficient estimates of the value $\theta$ of $x$ for which $M(\theta)$ is equal to some desired value. More importantly, these schemes make the "cost" of the observations, defined at the $n$th stage to be $\sum^n_1(x_i - \theta)^2$, to be of the order of $\log n$ instead of $n$, an obvious advantage in many applications. A general asymptotic theory is developed which includes these adaptive designs and the classical stochastic approximation schemes as special cases. Motivated by the cost considerations, some improvements are made in the pairwise sampling stochastic approximation scheme of Venter.

#### Article information

**Source**

Ann. Statist., Volume 7, Number 6 (1979), 1196-1221.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344840

**Digital Object Identifier**

doi:10.1214/aos/1176344840

**Mathematical Reviews number (MathSciNet)**

MR550144

**Zentralblatt MATH identifier**

0426.62059

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L20: Stochastic approximation

Secondary: 62K99: None of the above, but in this section 60F15: Strong theorems

**Keywords**

Adaptive design adaptive stochastic approximation regression logarithmic cost asymptotic normality iterated logarithm pairwise sampling schemes least squares

#### Citation

Lai, T. L.; Robbins, Herbert. Adaptive Design and Stochastic Approximation. Ann. Statist. 7 (1979), no. 6, 1196--1221. doi:10.1214/aos/1176344840. https://projecteuclid.org/euclid.aos/1176344840