, 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_specfun_erfcx_real (s15ag).

References

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

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: $x$ , the argument of the reciprocal of Mills' Ratio.

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.

Error Indicators and Warnings

None.

Accuracy

In the left-hand tail,

x < 0.0

, if

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

the safe range argument (nag_machine_real_safe (x02am)), then

0.0

is returned, which is close to the true value.

The relative accuracy is bounded by the effective machine precision. See nag_specfun_erfcx_real (s15ag) for further discussion.

Further Comments

If, before entry,

x

is not a standard Normal variable, it has to be standardized, and on exit, nag_stat_mills_ratio (g01mb) 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 .

Example

The hazard rate is evaluated at different values of

x

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

Open in the MATLAB editor: g01mb_example

function g01mb_example


fprintf('g01mb example results\n\n');

x    = [ 0.0; -2.0; 10.3];
xmu  = [ 0.0;  1.0;  9.0];
xsig = [ 1.0;  2.5;  1.6];

fprintf('  mean     sigma    x        reciprocal\n');
fprintf('                            Mills ratio\n\n');

for j = 1:numel(x)
  z  = (x(j)-xmu(j))/xsig(j);

  rm = g01mb(z);

  rm = rm/xsig(j);
  fprintf('%7.4f%9.4f%9.4f%11.4f\n', xmu(j), xsig(j), x(j), rm);
end

g01mb example results

  mean     sigma    x        reciprocal
                            Mills ratio

 0.0000   1.0000   0.0000     0.7979
 1.0000   2.5000  -2.0000     0.0878
 9.0000   1.6000  10.3000     0.8607

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox