Exact bounds on the inverse Mills ratio and its derivatives

Iosif Pinelis

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arXiv · 1512.00120

Exact bounds on the inverse Mills ratio and its derivatives

Iosif Pinelis

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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 $\Re z\ge0$ 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.

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BibTeX RIS

Iosif Pinelis. 2015-12-01. Exact bounds on the inverse Mills ratio and its derivatives. https://arxiv.org/abs/1512.00120

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