## The Annals of Statistics

### 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

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