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November 1997 Stein's method and the zero bias transformation with application to simple random sampling
Larry Goldstein, Gesine Reinert
Ann. Appl. Probab. 7(4): 935-952 (November 1997). DOI: 10.1214/aoap/1043862419

Abstract

Let W be a random variable with mean zero and variance $\sigma^2$. The distribution of a variate $W^*$, satisfying $EWf(W) = \sigma^2 Ef'(W^*)$ for smooth functions f , exists uniquely and defines the zero bias transformation on the distribution of W. The zero bias transformation shares many interesting properties with the well-known size bias transformation for nonnegative variables, but is applied to variables taking on both positive and negative values. The transformation can also be defined on more general random objects. The relation between the transformation and the expression $wf'(w) - \sigma^2 f''(w)$ which appears in the Stein equation characterizing the mean zero, variance $\sigma^2$ normal $\sigma Z$can be used to obtain bounds on the difference $E{h(W/ \sigma) - h(Z)}$ for smooth functions h by constructing the pair $(W, W^*)$ jointly on the same space. When W is a sum of n not necessarily independent variates, under certain conditions which include a vanishing third moment, bounds on this difference of the order $1/n$ for classes of smooth functions h may be obtained. The technique is illustrated by an application to simple random sampling.

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Larry Goldstein. Gesine Reinert. "Stein's method and the zero bias transformation with application to simple random sampling." Ann. Appl. Probab. 7 (4) 935 - 952, November 1997. https://doi.org/10.1214/aoap/1043862419

Information

Published: November 1997
First available in Project Euclid: 29 January 2003

zbMATH: 0903.60019
MathSciNet: MR1484792
Digital Object Identifier: 10.1214/aoap/1043862419

Subjects:
Primary: 60E10, 60F05, 62D05

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.7 • No. 4 • November 1997
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