Abstract:

Consider the Mills ratio $f(x) =(1-Phi(x))/phi(x)$, where $phi$  is the density function of the standard Gaussian law and $Phi$ its cumulative distribution. We introduce a general procedure to approximate $f$ for $x>0$ which allows to prove interesting properties where $f$ is involved. As applications we present a new proof that $1/f$ is strictly convex, and we give new sharp bounds of $f$ involving rational functions, functions with square roots or exponential terms. Also Chernoff type bounds for the Gaussian Q–function are studied. This is a joint work with F. Utzet.