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

Pingyi Fan|arXiv (Cornell University)|Dec 20, 2012

Mathematical Inequalities and Applications12 references22 citations

TL;DR

This paper presents two new inequalities for Mill's ratio that improve upon classical bounds by Gordon and Birnhaum-Sampford, offering tighter lower bounds for large x. The authors apply these to approximate the inverse Q-function, deriving simple, accurate expressions validated numerically, and propose a conjecture linking the inverse Q-function to information entropy for small probabilities.

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.

Motivation & Objective

• To derive tighter bounds for Mill's ratio to enhance Q-function approximation accuracy.
• To improve existing inequalities of Mill's ratio, particularly the lower bound, for large values of x.
• To develop practical approximations for the inverse Q-function using the new Mill's ratio bounds.
• To establish a theoretical and numerical foundation for approximating the inverse Q-function when the Q-function value is small.
• To propose a conjecture linking the inverse Q-function to information entropy for small α values.

Proposed method

• Derive a new upper bound for the Mill's ratio using the function $ g(u) = -1/√{1+u^2} $, leading to the inequality $ \int_x^\infty e^{-u^2/2} du < \frac{1}{\sqrt{1+x^2}} e^{-x^2/2} $ for $ x > \sqrt{(√{5}-1)/2} $.
• Establish a new lower bound using the monotonicity of $ g_3(u) = \frac{u(2+u^2)}{(1+u^2)^{3/2}} $, proving $ \frac{1+x^2}{x(2+x^2)} e^{-x^2/2} < \int_x^\infty e^{-u^2/2} du $ for $ x > \sqrt{2} $.
• Combine the new bounds with existing ones (Theorem 2) to form a piecewise tighter bound pair for all $ x > 0 $, improving accuracy over classical results.
• Use integration by parts and asymptotic analysis to derive approximations for the inverse Q-function, including $ Q^{-1}(\alpha) \approx \sqrt{ -\log(2\pi\alpha^2) } $ for small α.
• Propose a conjecture relating the inverse Q-function to the binary entropy function $ h(x) = -x\log x - (1-x)\log(1-x) $, suggesting $ Q^{-1}(\alpha) > \sqrt{ -\log h(2\pi\alpha^2) } $.
• Validate theoretical results through extensive numerical simulations comparing bounds and approximations against the exact integral and inverse Q-function.

Experimental results

Research questions

• RQ1 Can tighter inequalities for Mill's ratio be derived that improve upon the classical bounds of Gordon, Birnhaum, and Sampford?
• RQ2 How do the new bounds affect the accuracy of Q-function and inverse Q-function approximations, especially for large x or small α?
• RQ3 Can a simple, closed-form approximation for the inverse Q-function be derived using improved Mill's ratio bounds?
• RQ4 Is there a meaningful connection between the inverse Q-function and information entropy for small tail probabilities?
• RQ5 Can the proposed bounds and approximations be numerically validated and shown to converge rapidly as α → 0?

Key findings

• The new upper bound $ \frac{1}{\sqrt{1+x^2}} e^{-x^2/2} $ is valid for $ x > \sqrt{(√{5}-1)/2} \approx 0.7862 $, offering a simpler form than previous results.
• The new lower bound $ \frac{1+x^2}{x(2+x^2)} e^{-x^2/2} $ is tighter than the classical lower bound for $ x > \sqrt{2} $, especially for large x.
• Numerical results confirm that the new lower bound is tighter than the Birnhaum-Sampford lower bound for $ x > \sqrt{2} $, while the upper bound is looser but simpler.
• For $ x > 1.5 $, the new bounds provide better approximations than the classical ones, with $ \frac{1}{\sqrt{1+x^2}} e^{-x^2/2} $ being optimal among the tested bounds.
• The inverse Q-function approximation $ Q^{-1}(\alpha) \approx \sqrt{ -\log(2\pi\alpha^2) } $ performs well for small α, with numerical results showing rapid convergence.
• The conjecture $ Q^{-1}(\alpha) > \sqrt{ -\log h(2\pi\alpha^2) } $, where $ h(x) $ is the binary entropy function, is supported by numerical evidence for $ \alpha < 10^{-2} $, suggesting a deep link between entropy and tail probabilities.