New inequalities of Mill's ratio and Its Application to The Inverse Q-function Approximation

TL;DR

This paper introduces new, tighter inequalities for Mill's ratio and applies them to improve the approximation of the inverse Q-function, supported by numerical validation and a conjecture on inverse bounds.

Contribution

It presents novel, tighter inequalities for Mill's ratio and advances the inverse Q-function approximation with new theoretical results and conjectures.

Findings

New inequalities for Mill's ratio are tighter than existing ones.

Numerical results confirm the accuracy of the new inequalities.

A conjecture on bounds for the inverse Q-function is proposed.

Abstract

In this paper, we investigate the Mill's ratio estimation problem and get two new inequalities. Compared to the well known results obtained by Gordon, they becomes tighter. Furthermore, we also discuss the inverse Q-function approximation problem and present some useful results on the inverse solution. Numerical results confirm the validness of our theoretical analysis. In addition, we also present a conjecture on the bounds of inverse solution on Q-function.