# NAG FL Interfaceg05rkf (copula_​gumbel)

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## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine g05rkf ( n, m, x, ldx, sdx,
 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)
#include <nag.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)
The routine may be called by the names g05rkf or nagf_rand_copula_gumbel.

## 3Description

Generates $n$ pseudorandom uniform $m$-variates whose joint distribution is the Gumbel–Hougaard Archimedean copula ${C}_{\theta }$, given by
 $Cθ = exp{-[ (-ln⁡u1) θ + (-ln⁡u2) θ +⋯+ (-ln⁡um) θ ]} , { θ ∈ (1,∞) , uj ∈ (0,1] , j = 1 , 2 , … m ;$
with the special cases:
• ${C}_{1}={u}_{1}{u}_{2}\cdots {u}_{m}$, the product copula;
• ${C}_{\infty }=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({u}_{1},{u}_{2},\dots ,{u}_{m}\right)$, the Fréchet–Hoeffding upper bound.
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.

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of pseudorandom uniform variates to generate.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of dimensions.
Constraint: ${\mathbf{m}}\ge 2$.
3: $\mathbf{theta}$Real (Kind=nag_wp) Input
On entry: $\theta$, the copula parameter.
Constraint: ${\mathbf{theta}}\ge 1.0$.
4: $\mathbf{sorder}$Integer Input
On entry: 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 $\mathit{j}=1,2,\dots ,m$.
Constraint: ${\mathbf{sorder}}=1$ or $2$.
5: $\mathbf{state}\left(*\right)$Integer array Communication 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: $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{sdx}}\right)$Real (Kind=nag_wp) array Output
On exit: the pseudorandom uniform variates 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 of ${\mathbf{sorder}}=2$.
7: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g05rkf is called.
Constraints:
• if ${\mathbf{sorder}}=1$, ${\mathbf{ldx}}\ge {\mathbf{n}}$;
• if ${\mathbf{sorder}}=2$, ${\mathbf{ldx}}\ge {\mathbf{m}}$.
8: $\mathbf{sdx}$Integer Input
On entry: the second dimension of the array x as declared in the (sub)program from which g05rkf is called.
Constraints:
• if ${\mathbf{sorder}}=1$, ${\mathbf{sdx}}\ge {\mathbf{m}}$;
• if ${\mathbf{sorder}}=2$, ${\mathbf{sdx}}\ge {\mathbf{n}}$.
9: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, corrupt state argument.
${\mathbf{ifail}}=2$
On entry, invalid theta: ${\mathbf{theta}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{theta}}\ge 1.0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 2$.
${\mathbf{ifail}}=5$
On entry, invalid sorder.
Constraint: ${\mathbf{sorder}}=1$ or $2$.
${\mathbf{ifail}}=7$
On entry, ldx must be at least $⟨\mathit{\text{value}}⟩$: ${\mathbf{ldx}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=8$
On entry, sdx must be at least $⟨\mathit{\text{value}}⟩$: ${\mathbf{sdx}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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.

In practice, the need for numerical stability restricts the range of $\theta$ such that:
• if $\left(\theta -1\right)<1.0×{10}^{-6}$, the routine returns pseudorandom uniform variates with ${C}_{1}$ joint distribution;
• if $\theta >\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(80.0,-0.5\mathrm{ln}{\epsilon }_{s}\right)$, the routine returns pseudorandom uniform variates with ${C}_{\infty }$ joint distribution;
where ${\epsilon }_{s}$ is the safe-range parameter, the value of which is returned by x02amf.

## 10Example

This example generates thirteen four-dimensional variates for copula ${C}_{2.4}$.

### 10.1Program Text

Program Text (g05rkfe.f90)

### 10.2Program Data

Program Data (g05rkfe.d)

### 10.3Program Results

Program Results (g05rkfe.r)