Exact Bounds on the Inverse Mills Ratio and Its Derivatives

@article{Pinelis2015ExactBO,
  title={Exact Bounds on the Inverse Mills Ratio and Its Derivatives},
  author={Iosif Pinelis},
  journal={Complex Analysis and Operator Theory},
  year={2015},
  volume={13},
  pages={1643-1651}
}
  • I. Pinelis
  • Published 1 December 2015
  • Mathematics
  • Complex Analysis and Operator Theory
The inverse Mills ratio is $$R:=\varphi /\Psi $$R:=φ/Ψ, where $$\varphi $$φ and $$\Psi $$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 $$\mathfrak {R}z\geqslant 0$$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… Expand

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Complex Analysis