The Annals of Statistics

Nonlinear Stochastic Approximation Procedures for $L_p$ Loss Functions

Zhiliang Ying

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The classical stochastic approximation problem can be regarded as choosing design points so that the responses are close to some target level in the expected squared distance. Motivated by different loss criteria, a family of stochastic approximation algorithms is proposed. This family has the same simplicity as the classical Robbins-Monro procedure does and contains the latter as a special case. Using appropriate representations and martingale limit theorems, we establish asymptotic properties for this family. Using the semiparametric formulation, lower bounds are obtained for estimating the desired parameters under any adaptive design, showing that the proposed algorithms with appropriate scaling are asymptotically efficient.

Article information

Ann. Statist., Volume 18, Number 4 (1990), 1817-1828.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62L20: Stochastic approximation
Secondary: 62L05: Sequential design 60F17: Functional limit theorems; invariance principles

Stochastic approximation Robbins-Monro procedure sequential design $L_p$ loss information bound


Ying, Zhiliang. Nonlinear Stochastic Approximation Procedures for $L_p$ Loss Functions. Ann. Statist. 18 (1990), no. 4, 1817--1828. doi:10.1214/aos/1176347880.

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