The Annals of Statistics

Stochastic Approximation of an Implicity Defined Function

David Ruppert

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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.

Article information

Source
Ann. Statist., Volume 9, Number 3 (1981), 555-566.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345459

Digital Object Identifier
doi:10.1214/aos/1176345459

Mathematical Reviews number (MathSciNet)
MR615431

Zentralblatt MATH identifier
0476.62067

JSTOR
links.jstor.org

Subjects
Primary: 62L20: Stochastic approximation
Secondary: 62J99: None of the above, but in this section

Keywords
Stochastic approximation asymptotic normality process control

Citation

Ruppert, David. Stochastic Approximation of an Implicity Defined Function. Ann. Statist. 9 (1981), no. 3, 555--566. doi:10.1214/aos/1176345459. https://projecteuclid.org/euclid.aos/1176345459


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