naginterfaces.library.stat.mills_​ratio

naginterfaces.library.stat.mills_ratio(x)

mills_ratio returns the reciprocal of Mills’ Ratio.

For full information please refer to the NAG Library document for g01mb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01mbf.html

Parameters

xfloat

$x$, the argument of the reciprocal of Mills’ Ratio.

Returns

rmrfloat

The reciprocal of Mills’ Ratio.

Notes

mills_ratio calculates the reciprocal of Mills’ Ratio, the hazard rate, $\lambda \left(x\right)$, for the standard Normal distribution. It is defined as the ratio of the ordinate to the upper tail area of the standard Normal distribution, that is,

$$\lambda \left(x\right)=\frac{Z\left(x\right)}{Q\left(x\right)}=\frac{\frac{1}{\sqrt{2\pi }}{e}^{-\left(x^2/2\right)}}{\frac{1}{\sqrt{2\pi }}{\int }_x^{\infty }{e}^{-\left(t^2/2\right)}dt}\text{.}$$

The calculation is based on a Chebyshev expansion as described in specfun.erfcx_real.

References

Gross, A J and Clark, V A, 1975, Survival Distributions: Reliability Applications in the Biomedical Sciences, Wiley