$L^1$ bounds in normal approximation

Abstract

The zero bias distribution $W^{\ast }$ of $W$, defined though the characterizing equation $EWf(W)={\sigma }^{2}E{f}^{\prime }(W^{\ast })$ for all smooth functions $f$, exists for all $W$ with mean zero and finite variance ${\sigma }^2$. For $W$ and $W^{\ast }$ defined on the same probability space, the $L^1$ distance between $F$, the distribution function of $W$ with $EW=0$ and $Var(W)=1$, and the cumulative standard normal $\Phi$ has the simple upper bound \[\Vert F-Φ\Vert_1\le2E|W^*-W|.\] This inequality is used to provide explicit $L^1$ bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere $S({\ell }_n^p)$, simple random sampling and combinatorial central limit theorems.

Published in at http://dx.doi.org/10.1214/009117906000001123 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Stein's method and the zero bias transformation with application to simple random sampling
• Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing
• Normal approximation for hierarchical structures
• The central limit problem for random vectors with symmetries

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