Open Access
August 2007 A companion for the Kiefer–Wolfowitz–Blum stochastic approximation algorithm
Abdelkader Mokkadem, Mariane Pelletier
Ann. Statist. 35(4): 1749-1772 (August 2007). DOI: 10.1214/009053606000001451


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 $(\theta,\mu)$.


Download Citation

Abdelkader Mokkadem. Mariane Pelletier. "A companion for the Kiefer–Wolfowitz–Blum stochastic approximation algorithm." Ann. Statist. 35 (4) 1749 - 1772, August 2007.


Published: August 2007
First available in Project Euclid: 29 August 2007

zbMATH: 1209.62191
MathSciNet: MR2351104
Digital Object Identifier: 10.1214/009053606000001451

Primary: 62L20
Secondary: 62G08

Keywords: averaging principle , parametric rate , stochastic approximation algorithm , weak convergence rate

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 4 • August 2007
Back to Top