# NAG Library Routine Document

## 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)
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)

## 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.
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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of pseudorandom uniform variates to generate.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{m}$ – IntegerInput
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:     $\mathbf{sorder}$ – IntegerInput
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 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:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{sdx}}\right)$ – Real (Kind=nag_wp) arrayOutput
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:     $\mathbf{ldx}$ – IntegerInput
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:     $\mathbf{sdx}$ – IntegerInput
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:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism 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.

## 9Further Comments

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)

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