g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Documentnag_mills_ratio (g01mbc)

1  Purpose

nag_mills_ratio (g01mbc) returns the reciprocal of Mills' Ratio.

2  Specification

 #include #include
 double nag_mills_ratio (double x)

3  Description

nag_mills_ratio (g01mbc) 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 nag_erfcx (s15agc).

4  References

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

5  Arguments

1:     xdoubleInput
On entry: $x$, the argument of the reciprocal of Mills' Ratio.

None.

7  Accuracy

In the left-hand tail, $x<0.0$, if $\frac{1}{2}{e}^{-\left(1/2\right){x}^{2}}\le \text{}$ the safe range argument (nag_real_safe_small_number (X02AMC)), then $0.0$ is returned, which is close to the true value.
The relative accuracy is bounded by the effective machine precision. See nag_erfcx (s15agc) for further discussion.

If, before entry, $x$ is not a standard Normal variable, it has to be standardized, and on exit, nag_mills_ratio (g01mbc) 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
 $λx;μ,σ2=λx-μ/σ/σ.$

9  Example

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

9.1  Program Text

Program Text (g01mbce.c)

9.2  Program Data

Program Data (g01mbce.d)

9.3  Program Results

Program Results (g01mbce.r)