Exact Bounds on the Inverse Mills Ratio and Its Derivatives

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© 2018, Springer International Publishing AG, part of Springer Nature. 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.

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Complex Analysis and Operator Theory