Exact Bounds on the Inverse Mills Ratio and Its Derivatives
Document Type
Article
Publication Date
6-1-2019
Department
Department of Mathematical Sciences
Abstract
The inverse Mills ratio is R: = φ/ Ψ , where φ and Ψ are, respectively, the probability density function and the tail function of the standard normal distribution. Exact bounds on R(z) for complex z with Rz⩾ 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.
Publication Title
Complex Analysis and Operator Theory
Recommended Citation
Pinelis, I.
(2019).
Exact Bounds on the Inverse Mills Ratio and Its Derivatives.
Complex Analysis and Operator Theory,
13(4), 1643-1651.
http://doi.org/10.1007/s11785-018-0765-x
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/5093