标题： 标准正态累积分布及相关函数的波利亚上界改进 显示英文标题

标题： Improvements of Polya Upper Bound for Cumulative Standard Normal Distribution and Related Functions

摘要： 尽管关于累积标准正态分布的上界已有大量文献，但对于感兴趣的自变量 x 的所有值，这些上界相对不够精确。 本文的目的在于建立一个标准正态分布函数的精确上界，即其与 φ(x) 的最大绝对差值对于所有 x 值均小于某个值。 所确立的界改进了著名的波利亚（Polya）上界，并且可以作为 Φ(x) 本身的近似值，具有非常令人满意的确切度。 已经实现了所提出的上界与一些其他现有上界的数值比较，结果显示所提出的界比文献中找到的替代界更紧。 显示英文摘要

摘要： Although there is an extensive literature on the upper bound for cumulative standard normal distribution, there are relatively not sharp for all values of the interested argument x. The aim of this paper is to establish a sharp upper bound for standard normal distribution function, in the sense that its maximum absolute difference from phi(x) is less than for all values of x. The established bound improves the well-known Polya upper bound and it can be used as an approximation for Phi(x) itself with a very satisfactory accuracy. Numerical comparisons between the proposed upper bound and some other existing upper bounds have been achieved, which show that the proposed bound is tighter than alternative bounds found in the literature.