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September 2007 L1 bounds in normal approximation
Larry Goldstein
Ann. Probab. 35(5): 1888-1930 (September 2007). DOI: 10.1214/009117906000001123

Abstract

The zero bias distribution W* of W, defined though the characterizing equation EWf(W)=σ2Ef'(W*) for all smooth functions f, exists for all W with mean zero and finite variance σ2. For W and W* defined on the same probability space, the L1 distance between F, the distribution function of W with EW=0 and Var(W)=1, and the cumulative standard normal Φ has the simple upper bound

F−Φ‖1≤2E|W*W|.

This inequality is used to provide explicit L1 bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere S(np), simple random sampling and combinatorial central limit theorems.

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Larry Goldstein. "L1 bounds in normal approximation." Ann. Probab. 35 (5) 1888 - 1930, September 2007. https://doi.org/10.1214/009117906000001123

Information

Published: September 2007
First available in Project Euclid: 5 September 2007

MathSciNet: MR2349578
Digital Object Identifier: 10.1214/009117906000001123

Subjects:
Primary: 60C05 , 60D05 , 60F05 , 60F25

Keywords: Berry–Esseen , combinatorial CLT , cone measure , sampling , Stein’s method

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 5 • September 2007
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