NAG Library Function Document

nag_mills_ratio (g01mbc)

, 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 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: x – doubleInput: 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 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 Parallelism and Performance

Not applicable.

9 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

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 Function Documentnag_mills_ratio (g01mbc)

+− Contents

1 Purpose

2 Specification

3 Description