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_gumbel (g05rk).

References

Marshall A W and Olkin I (1988) Families of multivariate distributions Journal of the American Statistical Association 83 403

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 pseudorandom uniform variates to generate.

Constraint: $n \geq 0$ .
2: $m$ – int64int32nag_int scalar: $m$ , the number of dimensions.

Constraint: $m \geq 2$ .
3: $theta$ – double scalar: $θ$ , the copula parameter.

Constraint: $theta \geq 1.0$ .
4: $sorder$ – 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 $j = 1, 2, \dots, m$ .

Constraint: $sorder = 1$ or $2$ .
5: $state (:)$ – 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 pseudorandom uniform variates 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 of $sorder = 2$ .
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.

$ifail = 2$: Constraint: $theta \geq 1.0$ .

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

$ifail = 4$: Constraint: $m \geq 2$ .

$ifail = 5$: On entry, invalid sorder.
Constraint: $sorder = 1$ or $2$ .

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

$ifail = 8$: 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 $(θ - 1) < 1.0 \times 10^{- 6}$ , the function returns pseudorandom uniform variates with $C_{1}$ joint distribution;
if $θ > \max (80.0, - 0.5 \ln ε_{s})$ , 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).

Example

This example generates thirteen four-dimensional variates for copula

C_{2.4}

Open in the MATLAB editor: g05rk_example

function g05rk_example


fprintf('g05rk 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
n = int64(13);
m = int64(4);
% Sample order
sorder = int64(1);

% Parameter
theta = 2.4;

% Generate variates
[state, x, ifail] = g05rk( ...
                           n, m, theta, sorder, state);

disp('Variates from a Gumbel–Hougaard copula');
disp(x);

g05rk example results

Variates from a Gumbel–Hougaard copula
    0.9369    0.8676    0.9713    0.8854
    0.1139    0.3063    0.8625    0.2743
    0.4418    0.2211    0.5042    0.4985
    0.7902    0.6007    0.7493    0.6474
    0.8362    0.9847    0.8807    0.9079
    0.1781    0.4610    0.1283    0.1329
    0.1272    0.1760    0.1805    0.0383
    0.4473    0.2171    0.1662    0.1300
    0.8899    0.9005    0.8844    0.8879
    0.9069    0.8681    0.8450    0.8804
    0.2222    0.5499    0.4965    0.6488
    0.3807    0.5967    0.5096    0.3577
    0.8445    0.7755    0.8661    0.8948

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

Chapter Contents

Chapter Introduction

NAG Toolbox