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 $$\mathfrak {R}z\geqslant 0$$ are obtained, which then yield logarithmically exact upper bounds on high-order derivatives of R. These results complement the many known bounds on the (inverse) Mills ratio of the real argument. The main idea of the proof is a non-asymptotic version of the so-called stationary-phase method. This study was prompted by a recently discovered alternative to the Euler–Maclaurin formula.