NAG Library Routine Document

, 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,

λ (x) = \frac{Z (x)}{Q (x)} = \frac{\frac{1}{\sqrt{2 π}} e^{- (x^{2} / 2)}}{\frac{1}{\sqrt{2 π}} \int_{x}^{\infty} e^{- (t^{2} / 2)} d t} .

The calculation is based on a Chebyshev expansion as described in s15agf.

4

References

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

5

Arguments

1: $x$ – Real (Kind=nag_wp)Input: 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

\frac{1}{2} e^{- (1 / 2) x^{2}} \leq

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.

8

Parallelism and Performance

g01mbf is not threaded in any implementation.

9

Further Comments

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

μ

and variance

σ^{2}

, then its hazard rate,

λ (x; μ, σ^{2})

, is given by

λ (x; μ, σ^{2}) = λ ((x - μ) / σ) / σ .

10

Example

The hazard rate is evaluated at different values of

x

for Normal distributions with different means and variances. The results are then printed.

NAG Library Routine Document

g01mbf (mills_ratio)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description