NAG Library Function Document
nag_rand_sample (g05ndc)
1
Purpose
nag_rand_sample (g05ndc) selects a pseudorandom sample without replacement from an integer vector.
2
Specification
#include <nag.h> 
#include <nagg05.h> 
void 
nag_rand_sample (const Integer ipop[],
Integer n,
Integer isampl[],
Integer m,
Integer state[],
NagError *fail) 

3
Description
nag_rand_sample (g05ndc) selects
$m$ elements from a population vector
ipop of length
$n$ and places them in a sample vector
isampl. Their order in
ipop will be preserved in
isampl. Each of the
$\left(\begin{array}{c}n\\ m\end{array}\right)$ possible combinations of elements of
isampl may be regarded as being equally probable.
For moderate or large values of $n$ it is theoretically impossible that all combinations of size $m$ may occur, unless $m$ is near 1 or near $n$. This is because $\left(\begin{array}{c}n\\ m\end{array}\right)$ exceeds the cycle length of any of the base generators. For practical purposes this is irrelevant, as the time taken to generate all possible combinations is many millenia.
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 nonrepeatable sequence) must be called prior to the first call to
nag_rand_sample (g05ndc).
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:
$\mathbf{ipop}\left[{\mathbf{n}}\right]$ – const IntegerInput

On entry: the population to be sampled.
 2:
$\mathbf{n}$ – IntegerInput

On entry: the number of elements in the population vector to be sampled.
Constraint:
${\mathbf{n}}\ge 1$.
 3:
$\mathbf{isampl}\left[{\mathbf{m}}\right]$ – IntegerOutput

On exit: the selected sample.
 4:
$\mathbf{m}$ – IntegerInput

On entry: the sample size.
Constraint:
$1\le {\mathbf{m}}\le {\mathbf{n}}$.
 5:
$\mathbf{state}\left[\mathit{dim}\right]$ – IntegerCommunication Array

Note: the dimension,
$\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
state in the previous call to
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.
 6:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 NE_INT_2

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $1\le {\mathbf{m}}\le {\mathbf{n}}$.
 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_INVALID_STATE

On entry,
state vector has been corrupted or not initialized.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
Not applicable.
8
Parallelism and Performance
nag_rand_sample (g05ndc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken by nag_rand_sample (g05ndc) is of order $n$.
In order to sample other kinds of vectors, or matrices of higher dimension, the following technique may be used:
(a) 
set ${\mathbf{ipop}}\left[\mathit{i}1\right]=\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$; 
(b) 
use nag_rand_sample (g05ndc) to take a sample from ipop and put it into isampl; 
(c) 
use the contents of isampl as a set of indices to access the relevant vector or matrix. 
In order to divide a population into several groups,
nag_rand_permute (g05ncc) is more efficient.
10
Example
In the example program random samples of size
$1,2,\dots ,8$ are selected from a vector containing the first eight positive integers in ascending order. The samples are generated and printed for each sample size by a call to
nag_rand_sample (g05ndc) after initialization by
nag_rand_init_repeatable (g05kfc).
10.1
Program Text
Program Text (g05ndce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (g05ndce.r)