NAG Library Function Document

nag_rand_hypergeometric (g05tec)

+− 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_hypergeometric (g05tec) 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

k

with

m

specified items in it.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_hypergeometric (Nag_ModeRNG mode, Integer n, Integer ns, Integer np, Integer m, double r[], Integer lr, Integer state[], Integer x[], NagError *fail)

3 Description

nag_rand_hypergeometric (g05tec) generates

n

integers

x_{i}

from a discrete hypergeometric distribution, where the probability of

x_{i} = I

\begin{matrix} P (i = I) = \frac{l! m! (k - l)! (k - m)!}{I! (l - I)! (m - I)! (k - m - l + I)! k!} & if I = \max (0, m + l - k), \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_rand_hypergeometric (g05tec) 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 (k - l) (k - m) < 50 k^{3}

, otherwise Stirling's approximation is used.

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_hypergeometric (g05tec).

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_hypergeometric (g05tec).
$mode = Nag_InitializeAndGenerate$: 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: ns – IntegerInput

On entry:

l

, the sample size of the hypergeometric distribution.

Constraint:

0 \leq ns \leq np

4: np – IntegerInput

On entry:

k

, the population size of the hypergeometric distribution.

Constraint:

np \geq 0

5: m – IntegerInput

On entry:

m

, the number of specified items of the hypergeometric distribution.

Constraint:

0 \leq m \leq np

6: r[lr] – doubleCommunication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to nag_rand_hypergeometric (g05tec).

mode = Nag_GenerateWithoutReference

, r is not referenced by nag_rand_hypergeometric (g05tec).

On exit: the reference vector.

7: lr – IntegerInput

On entry: the dimension of the array r.

Suggested values:

if $mode \neq Nag_GenerateWithoutReference$ , $lr = 28 + 20 \times \sqrt{(ns \times m \times (np - m) \times (np - ns)) / {np}^{3}}$ approximately;
otherwise $lr = 1$ .

Constraints:

if $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ , lr must not be too small, but the limit is too complicated to specify;
if $mode = Nag_GenerateFromReference$ , lr must remain unchanged from the previous call to nag_rand_hypergeometric (g05tec).

8: 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.

9: x[n] – IntegerOutput

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

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, lr is too small when $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ : $lr = 〈value〉$ , minimum length required $= 〈value〉$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .; On entry, $np = 〈value〉$ .
Constraint: $np \geq 0$ .
NE_INT_2: On entry, $m = 〈value〉$ and $np = 〈value〉$ .
Constraint: $0 \leq m \leq np$ .; On entry, $ns = 〈value〉$ and $np = 〈value〉$ .
Constraint: $0 \leq ns \leq np$ .
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: The value of ns, np or m is not the same as when r was set up in a previous call with $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ .
NE_REF_VEC: On entry, some of the elements of the array r have been corrupted or have not been initialized.