Gaussian Mills ratio is completely monotone

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.