The Dvoretzky--Kiefer--Wolfowitz--Massart inequality gives a sub-Gaussian tail bound on the supremum norm distance between the empirical distribution function of a random sample and its population counterpart. We provide a short proof of a result that improves the existing bound in two respects. First, our one-sided bound holds without any restrictions on the failure probability, thereby verifying a conjecture of Birnbaum and McCarty (1958). Second, it is local in the sense that it holds uniformly over sub-intervals of the real line with an error rate that adapts to the behaviour of the population distribution function on the interval.

翻译：Dvoretzky–Kiefer–Wolfowitz–Massart 不等式给出了随机样本的经验分布函数与其总体对应函数之间上确界范数距离的亚高斯尾界。本文提供了一个简短证明，其结果在两个方面改进了现有界。首先，我们的单侧界对失败概率没有任何限制，从而验证了 Birnbaum 和 McCarty（1958）的一个猜想。其次，该界具有局部性，即它在实直线的子区间上一致成立，其误差率适应总体分布函数在该区间上的行为。