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A Bound for the Distribution of Smirnov’s Statistics
Lobachevskii Journal of Mathematics ( IF 0.8 ) Pub Date : 2021-04-18 , DOI: 10.1134/s1995080221020104 A. N. Doynikov , V. M. Kruglov
中文翻译:
斯米尔诺夫统计数据的界线
更新日期:2021-04-18
Lobachevskii Journal of Mathematics ( IF 0.8 ) Pub Date : 2021-04-18 , DOI: 10.1134/s1995080221020104 A. N. Doynikov , V. M. Kruglov
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:
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。