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On normal approximations to symmetric hypergeometric laws

Authors: Lutz Mattner and Jona Schulz
Journal: Trans. Amer. Math. Soc. 370 (2018), 727-748
MSC (2010): Primary 60E15; Secondary 60F05
Published electronically: September 7, 2017
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Abstract: The Kolmogorov distances between a symmetric hypergeometric law with standard deviation $ \sigma $ and its usual normal approximations are computed and shown to be less than $ 1/(\sqrt {8\pi }\,\sigma )$, with the order $ 1/\sigma $ and the constant $ 1/\sqrt {8\pi }$ being optimal. The results of Hipp and Mattner (2007) for symmetric binomial laws are obtained as special cases.

Connections to Berry-Esseen type results in more general situations concerning sums of simple random samples or Bernoulli convolutions are explained.

Auxiliary results of independent interest include rather sharp normal distribution function inequalities, a simple identifiability result for hypergeometric laws, and some remarks related to Lévy's concentration-variance inequality.

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Additional Information

Lutz Mattner
Affiliation: Universität Trier, Fachbereich IV – Mathematik, 54286 Trier, Germany

Jona Schulz
Affiliation: Universität Trier, Fachbereich IV – Mathematik, 54286 Trier, Germany

Keywords: Analytic inequalities, Bernoulli convolution, Berry-Esseen inequality, central limit theorem, concentration-variance inequality, finite population sampling, identifiability, normal distribution function inequalities, optimal error bound, remainder term estimate
Received by editor(s): April 24, 2014
Received by editor(s) in revised form: May 24, 2016
Published electronically: September 7, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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