void	nag_rand_copula_frank (Nag_OrderType order, Integer state[], double theta, Integer n, Integer m, double x[], Integer pdx, Integer sdx, NagError *fail)

3

Description

Generates

n

pseudorandom uniform

m

-variates whose joint distribution is the Frank Archimedean copula

C_{θ}

, given by

C_{θ} = - \frac{1}{θ} \ln [1 + \frac{(e^{- θ u_{1}} - 1) (e^{- θ u_{2}} - 1) \dots (e^{- θ u_{m}} - 1)}{{(e^{- θ} - 1)}^{m - 1}}], \{\begin{matrix} θ \in (0, \infty), \\ u_{j} \in (0, 1], j = 1, \dots m; \end{matrix}

with the special case:

$C_{\infty} = \min (u_{1}, u_{2}, \dots, u_{m})$ , the Fréchet–Hoeffding upper bound.

The generation method uses mixture of powers.

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_copula_frank (g05rjc).

4

References

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

5

Arguments

1: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $state [\dim]$ – IntegerCommunication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

3: $theta$ – doubleInput

On entry:

θ

, the copula parameter.

Constraint:

theta \geq 1.0 \times 10^{- 6}

4: $n$ – IntegerInput

On entry:

n

, the number of pseudorandom uniform variates to generate.

Constraint:

n \geq 0

5: $m$ – IntegerInput

On entry:

m

, the number of dimensions.

Constraint:

m \geq 2

6: $x [pdx \times sdx]$ – doubleOutput

Note: where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the pseudorandom uniform variates with joint distribution described by

C_{θ}

, with

X (i, j)

holding the

i

th value for the

j

th dimension if

order = Nag_ColMajor

and the

j

th value for the

i

th dimension of

order = Nag_RowMajor

7: $pdx$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

8: $sdx$ – IntegerInput

On entry: the secondary dimension of X.

Constraints:

if $order = Nag_ColMajor$ , $sdx \geq m$ ;
if $order = Nag_RowMajor$ , $sdx \geq n$ .

9: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 2$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, pdx must be at least $〈value〉$ : $pdx = 〈value〉$ .

On entry, sdx must be at least $〈value〉$ : $sdx = 〈value〉$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_STATE: On entry, corrupt state argument.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, invalid theta: $theta = 〈value〉$ .
Constraint: $theta \geq 1.0 \times 10^{- 6}$ .

7

Accuracy

Not applicable.

8

Parallelism and Performance

nag_rand_copula_frank (g05rjc) 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 function. 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:

the function requires $θ \geq 1.0 \times 10^{- 6}$ ;
if $θ > - \ln ε$ , the function returns pseudorandom uniform variates with $C_{\infty}$ joint distribution;

where

ε

is the machine precision returned by nag_machine_precision (X02AJC).

10

Example

This example generates thirteen four-dimensional variates for copula

C_{4.0}

10.1

Program Text

Program Text (g05rjce.c)

10.2

Program Data

None.

10.3

Program Results

Program Results (g05rjce.r)

nag_rand_copula_frank (g05rjc) (PDF version)

g05 Chapter Contents

g05 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_rand_copula_frank (g05rjc)

▸▿ 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

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