Exact bounds on the inverse Mills ratio and its derivatives

TL;DR

This paper derives precise bounds on the inverse Mills ratio and its derivatives for complex arguments, using a non-asymptotic stationary-phase method, which enhances understanding of its behavior in probability and statistical analysis.

Contribution

It provides the first exact bounds on the inverse Mills ratio and its derivatives for complex numbers with non-negative real parts, employing a novel non-asymptotic stationary-phase approach.

Findings

Exact bounds on the inverse Mills ratio for complex arguments.

Logarithmically precise bounds on derivatives of the ratio.

Introduction of a non-asymptotic stationary-phase method.

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.