NAG Library Function Document

nag_rngs_exp_mix (g05lqc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_rngs_exp_mix (g05lqc) generates a vector of pseudorandom numbers from an exponential mix distribution composed of

m

exponential distributions each having a mean

a_{i}

and weight

w_{i}

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rngs_exp_mix (Integer nmix, const double a[], const double wgt[], Integer n, double x[], Integer igen, Integer iseed[], NagError *fail)

3 Description

The distribution has PDF (probability density function)

f (x) = \{\begin{cases} \sum_{i = 1}^{m} \frac{1}{a_{i}} w_{i} e^{- x / a_{i}} & if ​ x > 0, \\ 0 & otherwise, \end{cases}

where

\sum_{i = 1}^{m} w_{i} = 1

and

a_{i} > 0

w_{i} \geq 0

nag_rngs_exp_mix (g05lqc) returns the values

x_{i}

by selecting, with probability

w_{j}

, random variates from an exponential distribution with parameter

a_{j}

One of the initialization functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_exp_mix (g05lqc).

4 References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

5 Arguments

1: nmix – IntegerInput

On entry:

m

, the number of exponential distributions in the mix.

Constraint:

nmix \geq 1

2: a[nmix] – const doubleInput

On entry: the

m

parameters

a_{i}

for the

m

exponential distributions in the mix.

Constraint:

a [i - 1] > 0.0

, for

i = 1, 2, \dots, nmix

3: wgt[nmix] – const doubleInput

On entry: the

m

weights

w_{i}

for the

m

exponential distributions in the mix.

Constraints:

$\sum_{i = 1}^{m} wgt [i - 1] = 1.0$ ;
$wgt [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, m$ .

4: n – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

5: x[n] – doubleOutput

On exit: the

n

pseudorandom numbers from the specified exponential mix distribution.

6: igen – IntegerInput

On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).

7: iseed[ $4$ ] – IntegerCommunication Array

On entry: contains values which define the current state of the selected generator.

On exit: contains updated values defining the new state of the selected generator.

8: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .; On entry, $nmix = 〈value〉$ .
Constraint: $nmix \geq 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL: On entry, sum of weights wgt is not equal to $1.0$ : sum $= 〈value〉$ .
NE_REAL_ARRAY_ELEM_CONS: On entry, $i = 〈value〉$ and $a [i - 1] = 〈value〉$ .
Constraint: $a [i - 1] > 0.0$ .; On entry, $i = 〈value〉$ and $wgt [i - 1] = 〈value〉$ .
Constraint: $wgt [i - 1] \geq 0.0$ .

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example prints the first five pseudorandom real numbers from an exponential mix distribution comprising three exponential distributions with parameters