In this article, we study bounds on the uniform distance between the cumulative distribution function of a standardized sum of independent centered random variables with moments of order four and its first-order Edgeworth expansion. Existing bounds are sharpened in two frameworks: when the variables are independent but not identically distributed and in the case of independent and identically distributed random variables. Improvements of these bounds are derived if the third moment of the distribution is zero. We also provide adapted versions of these bounds under additional regularity constraints on the tail behavior of the characteristic function. We finally present an application of our results to the lack of validity of one-sided tests based on the normal approximation of the mean for a fixed sample size.
翻译:在本文中,我们研究一个标准化的中央随机变量总和的累积分布功能与第4号命令时段及其第一级Edgeworth扩展之间的统一距离。现有的界限在两个框架中加以加强:当变量是独立的但非完全分布的,如果是独立和同样分布的随机变量。如果分布的第三个时刻是零,则可以对这些界限进行改进。我们还在特性函数尾部行为的额外规律性限制下提供这些界限的经调整的版本。我们最后根据固定样本大小平均值的正常近似值,对片面测试缺乏有效性的情况进行了应用。