nag_mills_ratio (g01mbc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_mills_ratio (g01mbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg01.h>
double  nag_mills_ratio (double x)

3  Description

nag_mills_ratio (g01mbc) calculates the reciprocal of Mills' Ratio, the hazard rate, λx, 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 12πxe-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.

6  Error Indicators and Warnings

None.

7  Accuracy

In the left-hand tail, x<0.0, if 12e-1/2x2 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.

8  Further Comments

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 μ and variance σ2, then its hazard rate, λx;μ,σ2, 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)


nag_mills_ratio (g01mbc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012