Some new upper and lower bounds for the Mills ratio

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.