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 highorder derivatives of $R$. The main idea of the proof is a nonasymptotic version of the socalled stationaryphase method.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1512.00120
 Bibcode:
 2015arXiv151200120P
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Probability;
 Primary 30A10;
 30D40;
 33B20;
 26D15;
 secondary 26A48;
 41A17;
 41A44;
 60E15;
 62E17
 EPrint:
 9 pages