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A Bound for the Distribution of Smirnov’s Statistics

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Abstract

Let \(F_{n}\) be the empirical distribution function for a sample of independent identically distributed random variables with distribution function \(F\). It is possible to prove the following inequality

$$\mathbb{P}\{\sqrt{n}\sup_{-\infty<x<\infty}(F_{n}(x)-F(x))>\lambda\}\leq c\exp\{-2\lambda^{2}-5\lambda^{4}/9n\},\quad\lambda\geq 0,$$

with some unspecified constant \(c\). There are some reasons based on numerous calculations to think that \(c=1\). It is shown in this paper that this hypothesis is true for \(2\leq n\leq 80\).

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Authors and Affiliations

  1. Department of Statistics, Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991, Moscow, Russia

    A. N. Doynikov & V. M. Kruglov

Authors
  1. A. N. Doynikov
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  2. V. M. Kruglov
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Correspondence to A. N. Doynikov or V. M. Kruglov.

Additional information

(Submitted by A. I. Volodin)

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Doynikov, A.N., Kruglov, V.M. A Bound for the Distribution of Smirnov’s Statistics. Lobachevskii J Math 42, 351–367 (2021). https://doi.org/10.1134/S1995080221020104

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  • DOI: https://doi.org/10.1134/S1995080221020104

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