NAG Library Routine Document

Integer, Intent (In)	::	n, 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	g05rff_ (const Integer n, const double theta, const Integer sorder, Integer state[], double x[], const Integer ldx, const Integer sdx, Integer ifail)

3

Description

Generates pseudorandom uniform bivariates

\{u_{1}, u_{2}\} \in {[0, 1]}^{2}

whose joint distribution is the Frank Archimedean copula

C_{θ}

with parameter

θ

, given by

C_{θ} = - \frac{1}{θ} \ln [1 + \frac{(e^{- θ u_{1}} - 1) (e^{- θ u_{2}} - 1)}{e^{- θ} - 1}], θ \in (- \infty, \infty) ∖ \{0\}

with the special cases:

$C_{- \infty} = \max (u_{1} + u_{2} - 1, 0)$ , the Fréchet–Hoeffding lower bound;
$C_{0} = u_{1} u_{2}$ , the product copula;
$C_{\infty} = \min (u_{1}, u_{2})$ , the Fréchet–Hoeffding upper bound.

The generation method uses conditional sampling.

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 g05rff.

4

References

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

5

Arguments

1: $n$ – IntegerInput

On entry:

n

, the number of bivariates to generate.

Constraint:

n \geq 0

2: $theta$ – Real (Kind=nag_wp)Input

On entry:

θ

, the copula parameter.

3: $sorder$ – IntegerInput

On entry: determines the storage order of variates; the

(i, j)

th variate is stored in

x (i, j)

sorder = 1

, and

x (j, i)

sorder = 2

, for

i = 1, 2, \dots, n

and

j = 1, 2

Constraint:

sorder = 1

2

4: $state (*)$ – Integer arrayCommunication Array

Note: the actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.

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.

5: $x (ldx, sdx)$ – Real (Kind=nag_wp) arrayOutput

On exit: the

n

bivariate uniforms 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 if

sorder = 2

6: $ldx$ – IntegerInput

On entry: the first dimension of the array x as declared in the (sub)program from which g05rff is called.

Constraints:

if $sorder = 1$ , $ldx \geq n$ ;
if $sorder = 2$ , $ldx \geq 2$ .

7: $sdx$ – IntegerInput

On entry: the second dimension of the array x as declared in the (sub)program from which g05rff is called.

Constraints:

if $sorder = 1$ , $sdx \geq 2$ ;
if $sorder = 2$ , $sdx \geq n$ .

8: $ifail$ – 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).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, corrupt state argument.

$ifail = 3$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

$ifail = 4$: On entry, invalid sorder.
Constraint: $sorder = 1$ or $2$ .

$ifail = 6$: On entry, ldx must be at least $〈value〉$ : $ldx = 〈value〉$ .

$ifail = 7$: On entry, sdx must be at least $〈value〉$ : $sdx = 〈value〉$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Not applicable.

8

Parallelism and Performance

g05rff 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.

9

Further Comments

In practice, the need for numerical stability restricts the range of

θ

such that:

if $θ < \ln ε_{s}$ , the routine returns pseudorandom uniform variates with $C_{- \infty}$ joint distribution;
if $|θ| < 1.0 \times 10^{- 6}$ , the routine returns pseudorandom uniform variates with $C_{0}$ joint distribution;
if $θ > \ln ε$ , the routine returns pseudorandom uniform variates with $C_{\infty}$ joint distribution;

where

ε_{s}

is the safe-range parameter, the value of which is returned by x02amf; and

ε

is the machine precision returned by x02ajf.

10

Example

This example generates thirteen variates for copula

C_{- 12.0}

NAG Library Routine Document

g05rff (copula_frank_bivar)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification