NAG Library Routine Document

1Purpose

g01mbf returns the reciprocal of Mills' Ratio.

2Specification

Fortran Interface
 Function g01mbf ( x)
 Real (Kind=nag_wp) :: g01mbf Real (Kind=nag_wp), Intent (In) :: x
#include <nagmk26.h>
 double g01mbf_ (const double *x)

3Description

g01mbf 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,
 $λx=Zx Qx =12πe-x2/2 12π∫x∞e-t2/2dt .$
The calculation is based on a Chebyshev expansion as described in s15agf.

4References

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

5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: $x$, the argument of the reciprocal of Mills' Ratio.

None.

7Accuracy

In the left-hand tail, $x<0.0$, if $\frac{1}{2}{e}^{-\left(1/2\right){x}^{2}}\le \text{}$ the safe range parameter (x02amf), then $0.0$ is returned, which is close to the true value.
The relative accuracy is bounded by the effective machine precision. See s15agf for further discussion.

8Parallelism and Performance

g01mbf is not threaded in any implementation.

If, before entry, $x$ is not a standard Normal variable, it has to be standardized, and on exit, g01mbf has to be divided by the standard deviation. That is, if the Normal distribution has mean $\mu$ and variance ${\sigma }^{2}$, then its hazard rate, $\lambda \left(x;\mu ,{\sigma }^{2}\right)$, is given by
 $λx;μ,σ2=λx-μ/σ/σ.$

10Example

The hazard rate is evaluated at different values of $x$ for Normal distributions with different means and variances. The results are then printed.

10.1Program Text

Program Text (g01mbfe.f90)

10.2Program Data

Program Data (g01mbfe.d)

10.3Program Results

Program Results (g01mbfe.r)