Gaussian Mills ratio is completely monotone

TL;DR

This paper proves that the Mills ratio of the standard Gaussian distribution is completely monotone, introduces rational bounds via continued fractions, and demonstrates that its reciprocal is strictly convex.

Contribution

It establishes the complete monotonicity of the Gaussian Mills ratio and provides a new approximation method using continued fractions.

Findings

Mills ratio is completely monotone.

Rational functions serve as sharp bounds and convergents.

The reciprocal of the Mills ratio is strictly convex.

Abstract

Consider the Mills ratio corresponding to the standard Gaussian law, $f(x)=\big(1-\Phi(x)\big)/\phi(x), \, x\ge 0$, where $\phi$ is the density function of this law and $\Phi$ its cumulative distribution function. We prove that this function is completely monotone. In the proof we obtain a sequence of rational functions that are sharp bounds for $f$; it turns out that these rational functions are the convergents of the continued fraction defined by $f$, and provide an approximation procedure that allows to prove interesting properties where $f$ or its derivatives are involved. As an application we show that $1/f$ is strictly convex.