Gaussian Mills ratio is completely monotone

Abstract

Consider the Mills ratio corresponding to the standard Gaussian law, $f(x)=(1-\Phi (x))/\varphi (x),x\ge 0$, where $\varphi$ 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.

This paper has been withdrawn by the authors due that the results appear in Arpad Baricz. Mills'ratio: Monotonicity patterns and functional inequalities. Journal of Mathematical Analysis and Applications, 340 (2008) 1362-1370 M. R. Sampford. Inequalities on Mill's ratio and related functions, The Annals of Mathematical Statistics, Vol. 24, No. 1 (1953) 130-132