NAG Library Routine Document

g05rkf  (copula_gumbel)


    1  Purpose
    7  Accuracy


g05rkf generates pseudorandom uniform variates with joint distribution of a Gumbel–Hougaard Archimedean copula.


Fortran Interface
Subroutine g05rkf ( n, m, theta, sorder, state, x, ldx, sdx, ifail)
Integer, Intent (In):: n, m, sorder, ldx, sdx
Integer, Intent (Inout):: state(*), ifail
Real (Kind=nag_wp), Intent (In):: theta
Real (Kind=nag_wp), Intent (Inout):: x(ldx,sdx)
C Header Interface
#include nagmk26.h
void  g05rkf_ ( const Integer *n, const Integer *m, const double *theta, const Integer *sorder, Integer state[], double x[], const Integer *ldx, const Integer *sdx, Integer *ifail)


Generates n pseudorandom uniform m-variates whose joint distribution is the Gumbel–Hougaard Archimedean copula Cθ, given by
Cθ = exp - -lnu1 θ + -lnu2 θ + + -lnum θ ,   θ 1, , uj 0,1 ,   j = 1 , 2 , m ;  
with the special cases:
The generation method uses mixture of powers.
One of the initialization routines g05kff (for a repeatable sequence if computed sequentially) or g05kgf (for a non-repeatable sequence) must be called prior to the first call to g05rkf.


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


1:     n – IntegerInput
On entry: n, the number of pseudorandom uniform variates to generate.
Constraint: n0.
2:     m – IntegerInput
On entry: m, the number of dimensions.
Constraint: m2.
3:     theta – Real (Kind=nag_wp)Input
On entry: θ, the copula parameter.
Constraint: theta1.0.
4:     sorder – IntegerInput
On entry: determines the storage order of variates; the i,jth variate is stored in xij if sorder=1, and xji if sorder=2, for i=1,2,,n and j=1,2,,m.
Constraint: sorder=1 or 2.
5:     state* – Integer arrayCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.
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:     xldxsdx – Real (Kind=nag_wp) arrayOutput
On exit: the pseudorandom uniform variates with joint distribution described by Cθ, with xij holding the ith value for the jth dimension if sorder=1 and the jth value for the ith dimension of sorder=2.
7:     ldx – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g05rkf is called.
  • if sorder=1, ldxn;
  • if sorder=2, ldxm.
8:     sdx – IntegerInput
On entry: the second dimension of the array x as declared in the (sub)program from which g05rkf is called.
  • if sorder=1, sdxm;
  • if sorder=2, sdxn.
9:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, corrupt state argument.
On entry, invalid theta: theta=value.
Constraint: theta1.0.
On entry, n=value.
Constraint: n0.
On entry, m=value.
Constraint: m2.
On entry, invalid sorder.
Constraint: sorder=1 or 2.
On entry, ldx must be at least value: ldx=value.
On entry, sdx must be at least value: sdx=value.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


Not applicable.

Parallelism and Performance

g05rkf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Further Comments

In practice, the need for numerical stability restricts the range of θ such that: where εs is the safe-range parameter, the value of which is returned by x02amf.


This example generates thirteen four-dimensional variates for copula C2.4.

Program Text

Program Text (g05rkfe.f90)

Program Data

Program Data (g05rkfe.d)

Program Results

Program Results (g05rkfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017