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May, 1981 Stochastic Approximation of an Implicity Defined Function
David Ruppert
Ann. Statist. 9(3): 555-566 (May, 1981). DOI: 10.1214/aos/1176345459

Abstract

Let $S$ be a set, $R$ the real line, and $M$ a real function on $R \times S$. Assume there exists a real function, $f$, on $S$ such that $(x - f(s))M(x, s) \geq 0$ for all $x$ and $s$. Initially neither $M$ nor $f$ are known. The goal is to estimate $f$. At time $n, s_n$ (a value in $S$) is observed, $x_n$ (a real number) is chosen, and an unbiased estimator of $M(x_n, s_n)$ is observed. This problem has applications, for example, to process control. In a previous paper the author proposed estimation of $f$ by a generalization of the Robbins-Monro procedure. Here that procedure is generalized and asymptotic distributions are studied.

Citation

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David Ruppert. "Stochastic Approximation of an Implicity Defined Function." Ann. Statist. 9 (3) 555 - 566, May, 1981. https://doi.org/10.1214/aos/1176345459

Information

Published: May, 1981
First available in Project Euclid: 12 April 2007

zbMATH: 0476.62067
MathSciNet: MR615431
Digital Object Identifier: 10.1214/aos/1176345459

Subjects:
Primary: 62L20
Secondary: 62J99

Keywords: asymptotic normality , process control , stochastic approximation

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 3 • May, 1981
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