## The Annals of Statistics

### A companion for the Kiefer–Wolfowitz–Blum stochastic approximation algorithm

#### Abstract

A stochastic algorithm for the recursive approximation of the location $θ$ of a maximum of a regression function was introduced by Kiefer and Wolfowitz [Ann. Math. Statist. 23 (1952) 462–466] in the univariate framework, and by Blum [Ann. Math. Statist. 25 (1954) 737–744] in the multivariate case. The aim of this paper is to provide a companion algorithm to the Kiefer–Wolfowitz–Blum algorithm, which allows one to simultaneously recursively approximate the size $μ$ of the maximum of the regression function. A precise study of the joint weak convergence rate of both algorithms is given; it turns out that, unlike the location of the maximum, the size of the maximum can be approximated by an algorithm which converges at the parametric rate. Moreover, averaging leads to an asymptotically efficient algorithm for the approximation of the couple $(\theta,\mu)$.

#### Article information

Source
Ann. Statist., Volume 35, Number 4 (2007), 1749-1772.

Dates
First available in Project Euclid: 29 August 2007

https://projecteuclid.org/euclid.aos/1188405629

Digital Object Identifier
doi:10.1214/009053606000001451

Mathematical Reviews number (MathSciNet)
MR2351104

Zentralblatt MATH identifier
1209.62191

Subjects
Primary: 62L20: Stochastic approximation
Secondary: 62G08: Nonparametric regression

#### Citation

Mokkadem, Abdelkader; Pelletier, Mariane. A companion for the Kiefer–Wolfowitz–Blum stochastic approximation algorithm. Ann. Statist. 35 (2007), no. 4, 1749--1772. doi:10.1214/009053606000001451. https://projecteuclid.org/euclid.aos/1188405629

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