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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_copula_frank_bivar (g05rf)

## Purpose

nag_rand_copula_frank_bivar (g05rf) generates pseudorandom uniform bivariates with joint distribution of a Frank Archimedean copula.

## Syntax

[state, x, ifail] = g05rf(n, theta, sorder, state)
[state, x, ifail] = nag_rand_copula_frank_bivar(n, theta, sorder, state)

## Description

Generates pseudorandom uniform bivariates $\left\{{u}_{1},{u}_{2}\right\}\in {\left[0,1\right]}^{2}$ whose joint distribution is the Frank Archimedean copula ${C}_{\theta }$ with parameter $\theta$, given by
 $Cθ = - 1θ ln 1 + e -θu1 -1 e -θu2 -1 e-θ-1 , θ ∈ -∞,∞ ∖ 0$
with the special cases:
• ${C}_{-\infty }=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({u}_{1}+{u}_{2}-1,0\right)$, the Fréchet–Hoeffding lower bound;
• ${C}_{0}={u}_{1}{u}_{2}$, the product copula;
• ${C}_{\infty }=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({u}_{1},{u}_{2}\right)$, the Fréchet–Hoeffding upper bound.
The generation method uses conditional sampling.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_copula_frank_bivar (g05rf).

## References

Nelsen R B (2006) An Introduction to Copulas (2nd Edition) Springer Series in Statistics

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of bivariates to generate.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{theta}$ – double scalar
$\theta$, the copula parameter.
3:     $\mathrm{sorder}$int64int32nag_int scalar
Determines the storage order of variates; the $\left(\mathit{i},\mathit{j}\right)$th variate is stored in ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ if ${\mathbf{sorder}}=1$, and ${\mathbf{x}}\left(\mathit{j},\mathit{i}\right)$ if ${\mathbf{sorder}}=2$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2$.
Constraint: ${\mathbf{sorder}}=1$ or $2$.
4:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

None.

### Output Parameters

1:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
2:     $\mathrm{x}\left(\mathit{ldx},\mathit{sdx}\right)$ – double array
The $n$ bivariate uniforms with joint distribution described by ${C}_{\theta }$, with ${\mathbf{x}}\left(i,j\right)$ holding the $i$th value for the $j$th dimension if ${\mathbf{sorder}}=1$ and the $j$th value for the $i$th dimension if ${\mathbf{sorder}}=2$.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
On entry, corrupt state argument.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=4$
On entry, invalid sorder.
Constraint: ${\mathbf{sorder}}=1$ or $2$.
${\mathbf{ifail}}=6$
On entry, ldx is too small: $\mathit{ldx}=_$.
${\mathbf{ifail}}=7$
On entry, sdx is too small: $\mathit{sdx}=_$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

In practice, the need for numerical stability restricts the range of $\theta$ such that:
• if $\theta <\mathrm{ln}{\epsilon }_{s}$, the function returns pseudorandom uniform variates with ${C}_{-\infty }$ joint distribution;
• if $\left|\theta \right|<1.0×{10}^{-6}$, the function returns pseudorandom uniform variates with ${C}_{0}$ joint distribution;
• if $\theta >\mathrm{ln}\epsilon$, the function returns pseudorandom uniform variates with ${C}_{\infty }$ joint distribution;
where ${\epsilon }_{s}$ is the safe-range parameter, the value of which is returned by nag_machine_real_safe (x02am); and $\epsilon$ is the machine precision returned by nag_machine_precision (x02aj).

## Example

This example generates thirteen variates for copula ${C}_{-12.0}$.
```function g05rf_example

fprintf('g05rf example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
genid, subid, seed);

% Sample size and order
n      = int64(13);
sorder = int64(1);

% Parameter
theta = -12;

% Generate variates
[state, x, ifail] = g05rf( ...
n, theta, sorder, state);

disp('Variates from a bivariate Frank copula');
disp(x);

```
```g05rf example results

Variates from a bivariate Frank copula
0.6364    0.1411
0.1065    0.8967
0.7460    0.1843
0.7983    0.1254
0.1046    0.9982
0.4925    0.6901
0.3843    0.6250
0.7871    0.1654
0.4982    0.5298
0.6717    0.2902
0.0505    0.9554
0.2580    0.8190
0.6238    0.3014

```