The Annals of Applied Probability

Stein's method and the zero bias transformation with application to simple random sampling

Larry Goldstein and Gesine Reinert

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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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Ann. Appl. Probab., Volume 7, Number 4 (1997), 935-952.

First available in Project Euclid: 29 January 2003

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 62D05: Sampling theory, sample surveys 60E10: Characteristic functions; other transforms

Normal approximation Stein's method simple random sampling size biasing distributional transformation coupling


Goldstein, Larry; Reinert, Gesine. Stein's method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 (1997), no. 4, 935--952. doi:10.1214/aoap/1043862419.

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