## The Annals of Statistics

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

### Stochastic Approximation of an Implicity Defined Function

#### 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