The Annals of Statistics

Adaptive Design and Stochastic Approximation

T. L. Lai and Herbert Robbins

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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

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

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62L20: Stochastic approximation
Secondary: 62K99: None of the above, but in this section 60F15: Strong theorems

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


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

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