NAG Library Function Document

nag_rand_neg_bin (g05thc)

+− 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_rand_neg_bin (g05thc) 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_rand_neg_bin (Nag_ModeRNG mode, Integer n, Integer m, double p, double r[], Integer lr, Integer state[], Integer x[], NagError *fail)

3 Description

nag_rand_neg_bin (g05thc) 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_rand_neg_bin (g05thc) with the same parameter value can then use this reference vector to generate further variates.

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_neg_bin (g05thc).

4 References

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

5 Arguments

1: mode – Nag_ModeRNGInput

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

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate variates using reference vector set up in a prior call to nag_rand_neg_bin (g05thc).
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.
$mode = Nag_GenerateWithoutReference$: Generate variates without using the reference vector.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

Nag_GenerateWithoutReference

2: n – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

3: m – IntegerInput

On entry:

m

, the number of failures of the distribution.

Constraint:

m \geq 0

4: 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

5: r[lr] – doubleCommunication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to nag_rand_neg_bin (g05thc).

mode = Nag_GenerateWithoutReference

, r is not referenced by nag_rand_neg_bin (g05thc).

On exit: the reference vector.

6: lr – IntegerInput

On entry: the dimension of the array r.

Suggested values:

if $mode \neq Nag_GenerateWithoutReference$ ,
$lr = 28 + (20 \times \sqrt{m \times p} + 30 \times p) / (1 - p) approximately$ ;
otherwise $lr = 1$ .

Constraints:

if $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ ,
$\begin{matrix} lr > & int (\begin{matrix} \frac{m \times p + 7.15 \times \sqrt{m \times p} + 20.15 \times p}{1 - p} + 8.5 \end{matrix}) \\ - \max (\begin{matrix} 0, int (\begin{matrix} \frac{m \times p - 7.15 \times \sqrt{m \times p}}{1 - p} \end{matrix}) \end{matrix}) + 9 \end{matrix}$ ;
if $mode = Nag_GenerateFromReference$ , lr must remain unchanged from the previous call to nag_rand_neg_bin (g05thc).

7: state[ $\dim$ ] – IntegerCommunication Array

Note: the actual argument supplied must be the array state supplied to the initialization functions nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

8: x[n] – IntegerOutput

On exit: the

n

pseudorandom numbers from the specified negative binomial distribution.

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_INT: On entry, lr is too small when $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ : $lr = 〈value〉$ , minimum length required $= 〈value〉$ .; On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
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, $p = 〈value〉$ .
Constraint: $0.0 \leq p < 1.0$ .
NE_REF_VEC: On entry, some of the elements of the array r have been corrupted or have not been initialized.