Abstract In this paper, we present new upper and lower bounds for the Mills ratio of the standard Gaussian law. Several different methods are used to derive these new bounds. One of the methods reproduces the bounds of several different authors in previous works as special cases and is a very general method that produces many new bounds. One of the bounds can be written in terms of hyperbolic sine and inverse hyperbolic sine functions. Some of the bounds involve exponential functions and are improved versions of previously proposed bounds or are improved versions of the new bounds introduced earlier in this paper. Some results from reliability theory and Jensen's inequality are used to improve determinantal inequalities. Some open problems are discussed, and conjectures are made.

中文翻译：

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米尔斯比率的一些新上限和下限

摘要 在本文中，我们提出了标准高斯定律的米尔斯比的新上限和下限。使用几种不同的方法来推导这些新的界限。其中一种方法将以前作品中几个不同作者的边界作为特例复制，并且是一种非常通用的方法，可以产生许多新的边界。边界之一可以写成双曲正弦函数和反双曲正弦函数。一些边界涉及指数函数，是先前提出的边界的改进版本，或者是本文前面介绍的新边界的改进版本。可靠性理论和 Jensen 不等式的一些结果用于改进行列式不等式。讨论了一些悬而未决的问题，并进行了猜想。