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December, 1990 Nonlinear Stochastic Approximation Procedures for $L_p$ Loss Functions
Zhiliang Ying
Ann. Statist. 18(4): 1817-1828 (December, 1990). DOI: 10.1214/aos/1176347880


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.


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Zhiliang Ying. "Nonlinear Stochastic Approximation Procedures for $L_p$ Loss Functions." Ann. Statist. 18 (4) 1817 - 1828, December, 1990.


Published: December, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0716.62080
MathSciNet: MR1074437
Digital Object Identifier: 10.1214/aos/1176347880

Primary: 62L20
Secondary: 60F17 , 62L05

Keywords: $L_p$ loss , information bound , Robbins-Monro procedure , sequential design , stochastic approximation

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 4 • December, 1990
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