Exact bounds on the inverse Mills ratio and its derivatives

Abstract

The inverse Mills ratio is $R:=\varphi/\Psi$, where $\varphi$ and $\Psi$ are, respectively, the probability density function and the tail function of the standard normal distribution. Exact bounds on $R(z)$ for complex $z$ with $\Re z\ge0$ are obtained, which then yield logarithmically exact bounds on high-order derivatives of $R$. The main idea of the proof is a non-asymptotic version of the so-called stationary-phase method.