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NAG Toolbox: nag_rand_copula_clayton_bivar (g05re)

Purpose

nag_rand_copula_clayton_bivar (g05re) generates pseudorandom uniform bivariates with joint distribution of a Clayton/Cook–Johnson Archimedean copula.

Syntax

[state, x, ifail] = g05re(n, theta, sorder, state)
[state, x, ifail] = nag_rand_copula_clayton_bivar(n, theta, sorder, state)

Description

Generates pseudorandom uniform bivariates {u1,u2}(0,1]2{u1,u2}(0,1]2 whose joint distribution is the Clayton/Cook–Johnson Archimedean copula CθCθ with parameter θθ, given by
Cθ = [max ( u1θ + u2θ 1 ,0)]1 / θ ,   θ (1,) {0}
Cθ = [ max( u1 -θ + u2 -θ -1 ,0) ] -1/θ ,   θ (-1,) {0}
with the special cases:
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_clayton_bivar (g05re).

References

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

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
nn, the number of bivariates to generate.
Constraint: n0n0.
2:     theta – double scalar
θθ, the copula parameter.
Constraint: theta1.0theta-1.0.
3:     sorder – int64int32nag_int scalar
Determines the storage order of variates; the (i,j)(i,j)th variate is stored in x(i,j)xij if sorder = 1sorder=1, and x(j,i)xji if sorder = 2sorder=2, for i = 1,2,,ni=1,2,,n and j = 1,2j=1,2.
Constraint: sorder = 1sorder=1 or 22.
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.

Input Parameters Omitted from the MATLAB Interface

ldx sdx

Output Parameters

1:     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 updated information on the state of the generator.
2:     x(ldx,sdx) – double array
The nn bivariate uniforms with joint distribution described by CθCθ, with x(i,j)xij holding the iith value for the jjth dimension if sorder = 1sorder=1 and the jjth value for the iith dimension of sorder = 2sorder=2.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry, corrupt state parameter.
  ifail = 2ifail=2
Constraint: theta1.0theta-1.0.
  ifail = 3ifail=3
Constraint: n0n0.
  ifail = 4ifail=4
On entry, invalid sorder.
Constraint: sorder = 1sorder=1 or 22.
  ifail = 6ifail=6
On entry, ldx is too small: .
  ifail = 7ifail=7
On entry, sdx is too small: .
  ifail = 999ifail=-999
Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

In practice, the need for numerical stability restricts the range of θθ such that: where εsεs is the safe-range parameter, the value of which is returned by nag_machine_real_safe (x02am); and εε is the machine precision returned by nag_machine_precision (x02aj).

Example

function nag_rand_copula_clayton_bivar_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
% Sample size
n = int64(13);
% Sample order
sorder = int64(1);
% Parameter
theta = -0.8;

% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);

% Generate variates
[state, x, ifail] = nag_rand_copula_clayton_bivar(n, theta, sorder, state)
 

state =

                   17
                 1234
                    1
                    0
                28214
                15039
                27035
                23461
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    0.6400    0.2223
    0.1154    0.8101
    0.7486    0.1439
    0.8003    0.1062
    0.1135    0.9946
    0.4975    0.7655
    0.3904    0.4925
    0.7892    0.1196
    0.5032    0.4116
    0.6750    0.2093
    0.0600    0.9055
    0.2655    0.7085
    0.6276    0.2370


ifail =

                    0


function g05re_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
% Sample size
n = int64(13);
% Sample order
sorder = int64(1);
% Parameter
theta = -0.8;

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

% Generate variates
[state, x, ifail] = g05re(n, theta, sorder, state)
 

state =

                   17
                 1234
                    1
                    0
                28214
                15039
                27035
                23461
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    0.6400    0.2223
    0.1154    0.8101
    0.7486    0.1439
    0.8003    0.1062
    0.1135    0.9946
    0.4975    0.7655
    0.3904    0.4925
    0.7892    0.1196
    0.5032    0.4116
    0.6750    0.2093
    0.0600    0.9055
    0.2655    0.7085
    0.6276    0.2370


ifail =

                    0



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