The Annals of Statistics

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

Abdelkader Mokkadem and Mariane Pelletier

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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

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

First available in Project Euclid: 29 August 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

stochastic approximation algorithm weak convergence rate parametric rate averaging principle


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.

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