NAG Library Function Document

nag_rngs_neg_bin (g05mcc)

+− 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_neg_bin (g05mcc) generates a vector of pseudorandom integers from the discrete negative binomial distribution with parameter

m

and probability

p

of success at a trial.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rngs_neg_bin (Integer mode, Integer m, double p, Integer n, Integer x[], Integer igen, Integer iseed[], double r[], NagError *fail)

3 Description

nag_rngs_neg_bin (g05mcc) generates

n

integers

x_{i}

from a discrete negative binomial distribution, where the probability of

x_{i} = I

(

I

successes before

m

failures) is

P (x_{i} = I) = \frac{(m + I - 1)!}{I! (m - 1)!} \times p^{I} \times {(1 - p)}^{m}, I = 0, 1, \dots .

The variates can be generated with or without using a search table and index. If a search table is used then it is stored with the index in a reference vector and subsequent calls to nag_rngs_neg_bin (g05mcc) with the same parameter value can then use this reference vector to generate further variates.

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_neg_bin (g05mcc).

4 References

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

5 Arguments

1: mode – IntegerInput

On entry: a code for selecting the operation to be performed by the function.

$mode = 0$: Set up reference vector only.
$mode = 1$: Generate variates using reference vector set up in a prior call to nag_rngs_neg_bin (g05mcc).
$mode = 2$: Set up reference vector and generate variates.
$mode = 3$: Generate variates without using the reference vector.

Constraint:

mode = 0

1

2

3

2: m – IntegerInput

On entry:

m

, the number of failures of the distribution.

Constraint:

m \geq 0

3: p – doubleInput

On entry:

p

, the parameter of the negative binomial distribution representing the probability of success at a single trial.

Constraint:

0.0 \leq p < 1.0

4: n – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 1

5: x[n] – IntegerOutput

On exit: the

n

pseudorandom numbers from the specified negative binomial 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: r[ $\dim$ ] – doubleCommunication Array

Note: the dimension, dim, of the array r must be at least

$20 + (20 \sqrt{m \times p} + 30 \times p) / (1 - p)$ when $mode \neq 3$ ;
$1$ otherwise.

On entry: if

mode = 1

, the reference vector from the previous call to nag_rngs_neg_bin (g05mcc).

On exit: the reference vector.

9: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_DIM_INFEASIBLE: p and m are such that the reference vector length would exceed integer range. We recommend setting $mode = 3$ .
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .; On entry, $mode = 〈value〉$ .
Constraint: $mode = 0$ , $1$ , $2$ or $3$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
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_PREV_CALL: p or m is not the same as when r was set up in a previous call. Previous value of $p = 〈value〉$ and $p = 〈value〉$ . Previous value of $m = 〈value〉$ and $m = 〈value〉$ .
NE_REAL: On entry, $p < 0.0$ or $p \geq 1.0$ : $p = 〈value〉$ .