NAG Library Function Document

nag_rngs_hypergeometric (g05mlc)

+− 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_hypergeometric (g05mlc) generates a vector of pseudorandom integers from the discrete hypergeometric distribution of the number of specified items in a sample of size

l

, taken from a population of size

n

with

m

specified items in it.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rngs_hypergeometric (Integer mode, Integer ns, Integer np, Integer m, Integer n, Integer x[], Integer igen, Integer iseed[], double r[], NagError *fail)

3 Description

nag_rngs_hypergeometric (g05mlc) generates a vector of integers,

x_{i}

, from a discrete hypergeometric distribution, where the probability of

x_{i} = I

\begin{matrix} P (i = I) = \frac{l! m! (n - l)! (n - m)!}{I! (l - I)! (m - I)! (n - m - l + I)! n!} & if I = \max (0, m + l - n), \dots, \min (l, m), \\ P (i = I) = 0 & otherwise. \end{matrix}

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_hypergeometric (g05mlc) with the same parameter values can then use this reference vector to generate further variates. The reference array is generated by a recurrence relation if

l m (n - l) (n - m) < 50 n^{3}

, otherwise Stirling's approximation is used.

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_hypergeometric (g05mlc).

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_hypergeometric (g05mlc).
$mode = 2$: Set up reference vector and generate variates.
$mode = 3$: Generate variates without using the reference vector.

Constraint:

mode = 0

1

2

3

2: ns – IntegerInput

On entry:

l

, the sample size of the hypergeometric distribution.

Constraint:

0 \leq ns \leq np

3: np – IntegerInput

On entry:

n

, the population size of the hypergeometric distribution.

Constraint:

np \geq 0

4: m – IntegerInput

On entry:

m

, the number of specified items of the hypergeometric distribution.

Constraint:

0 \leq m \leq np

5: n – IntegerInput

On entry: the number of pseudorandom numbers to be generated.

Constraint:

n \geq 1

6: x[n] – IntegerOutput

On exit: the pseudorandom numbers from the specified hypergeometric distribution.

7: 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).

8: 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.

9: r[ $\dim$ ] – doubleCommunication Array

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

20 + 20 \times \sqrt{(ns \times m \times (np - m) \times (np - ns)) / {np}^{3}}

On entry: if

mode = 1

, the reference vector from the previous call to nag_rngs_hypergeometric (g05mlc).

On exit: the reference vector.

10: 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, $mode = 〈value〉$ .
Constraint: $mode = 0$ , $1$ , $2$ or $3$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .; On entry, $np = 〈value〉$ .
Constraint: $np \geq 0$ .
NE_INT_2: On entry, $m = 〈value〉$ and $np = 〈value〉$ .
Constraint: $m \leq np$ and $m \geq 0$ .; On entry, $ns = 〈value〉$ and $np = 〈value〉$ .
Constraint: $ns \leq np$ and $ns \geq 0$ .
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_PREV_CALL: ns or np or m is not the same as when r was set up in a previous call or the data in r has been corrupted.