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 \(2\leq n\leq 80\).

中文翻译：

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斯米尔诺夫统计数据的界线

摘要

令\（F_ {n} \）为具有分布函数\（F \）的独立同分布随机变量样本的经验分布函数。可以证明以下不等式

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

带有一些未指定的常量\（c \）。基于许多计算，有一些理由认为\（c = 1 \）。本文证明了这个假设对于\（2 \ leq n \ leq 80 \）是正确的。