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November, 1973 Transformation of Observations in Stochastic Approximation
Sami Naguib Abdelhamid
Ann. Statist. 1(6): 1158-1174 (November, 1973). DOI: 10.1214/aos/1176342564

Abstract

The general stochastic approximation procedure: $X_{n+1} = X_n - a_n c_n^{-1}h(Y_n),\quad n = 1,2, \cdots$ is considered, where $h$ is a Borel measurable transformation on the random observations $Y_n$. Under some mild requirements on $h$ and on the error random variables, the asymptotic properties, the a.s. convergence and the asymptotic normality are studied. The analysis is confined to the case where the error random variables are (conditionally) distributed according to a distribution function $G$ which is symmetric around 0 and admits a density $g$. The optimal choices of the design sequences $a_n$ and $c_n$ as well as the transformation $h$ are studied. The optimal transformation turned out to be equal to $-C(g'/g)$ (a.e. with respect to $G$) for a $C > 0$ and it is the transformation which minimizes the second moment of the asymptotic distribution of $n^\beta(X_n - \theta)$. The Robbins-Monro and the Kiefer-Wolfowitz situations are emphasized as special cases. With the optimal transformation, the new proposed generalized procedure is shown to yield asymptotically efficient estimators.

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Sami Naguib Abdelhamid. "Transformation of Observations in Stochastic Approximation." Ann. Statist. 1 (6) 1158 - 1174, November, 1973. https://doi.org/10.1214/aos/1176342564

Information

Published: November, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0303.62065
MathSciNet: MR348944
Digital Object Identifier: 10.1214/aos/1176342564

Keywords: asymptotically efficient estimators , Kiefer-Wolfowitz situation , optimal procedure , optimal transformation , Robbins-Monro situation , stochastic approximation procedure , Transformation of observations

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 6 • November, 1973
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