Approximating Mills ratio

TL;DR

This paper introduces a new general method to approximate the Mills ratio across its entire domain, enabling proofs of properties and deriving sharp bounds, with applications to Gaussian tail probability bounds.

Contribution

A novel approximation procedure for the Mills ratio that facilitates property proofs and sharp bounds, including Chernoff bounds for Gaussian functions.

Findings

Proved that 1/f is strictly convex.

Derived new sharp bounds involving rational, square root, and exponential functions.

Studied Chernoff type bounds for the Gaussian Q-function.

Abstract

Consider the Mills ratio $f(x)=\big(1-\Phi(x)\big)/\phi(x), \, x\ge 0$, where $\phi$ is the density function of the standard Gaussian law and $\Phi$ its cumulative distribution.We introduce a general procedure to approximate $f$ on the whole $[0,\infty)$ which allows to prove interesting properties where $f$ is involved. As applications we present a new proof that $1/f$ is strictly convex, and we give new sharp bounds of $f$ involving rational functions, functions with square roots or exponential terms. Also Chernoff type bounds for the Gaussian $Q$--function are studied.