On normal approximations to symmetric hypergeometric laws

Mattner, Lutz; Schulz, Jona

doi:10.1090/tran/6986

On normal approximations to symmetric hypergeometric laws
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by Lutz Mattner and Jona Schulz PDF

Trans. Amer. Math. Soc. 370 (2018), 727-748 Request permission

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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