$C_{- \infty} = \max (u_{1} + u_{2} - 1, 0)$ , the Fréchet–Hoeffding lower bound;
$C_{0} = u_{1} u_{2}$ , the product copula;
$C_{\infty} = \min (u_{1}, u_{2})$ , 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: $n$ – int64int32nag_int scalar: $n$ , the number of bivariates to generate.

Constraint: $n \geq 0$ .
2: $theta$ – double scalar: $θ$ , the copula parameter.
3: $sorder$ – int64int32nag_int scalar: Determines the storage order of variates; the $(i, j)$ th variate is stored in $x (i, j)$ if $sorder = 1$ , and $x (j, i)$ if $sorder = 2$ , for $i = 1, 2, \dots, n$ and $j = 1, 2$ .

Constraint: $sorder = 1$ or $2$ .
4: $state (:)$ – 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.

Optional Input Parameters

None.

Output Parameters

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

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, corrupt state argument.

$ifail = 3$: Constraint: $n \geq 0$ .

$ifail = 4$: On entry, invalid sorder.
Constraint: $sorder = 1$ or $2$ .

$ifail = 6$: On entry, ldx is too small: $ldx =_$ .

$ifail = 7$: On entry, sdx is too small: $sdx =_$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

In practice, the need for numerical stability restricts the range of

θ

such that:

if $θ < \ln ε_{s}$ , the function returns pseudorandom uniform variates with $C_{- \infty}$ joint distribution;
if $|θ| < 1.0 \times 10^{- 6}$ , the function returns pseudorandom uniform variates with $C_{0}$ joint distribution;
if $θ > \ln ε$ , the function returns pseudorandom uniform variates with $C_{\infty}$ joint distribution;

where

ε_{s}

is the safe-range parameter, the value of which is returned by nag_machine_real_safe (x02am); and

ε

is the machine precision returned by nag_machine_precision (x02aj).

Example

This example generates thirteen variates for copula

C_{- 12.0}

Open in the MATLAB editor: g05rf_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox