NAG Library Function Document

nag_rngs_copula_normal (g05rac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_rngs_copula_normal (g05rac) sets up a reference vector and generates an array of pseudorandom numbers from a Normal (Gaussian) copula with covariance matrix

C

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rngs_copula_normal (Nag_OrderType order, Integer mode, Integer m, const double c[], Integer pdc, Integer n, double x[], Integer pdx, Integer igen, Integer iseed[], double r[], Integer lr, NagError *fail)

3 Description

The Gaussian copula,

c

, is defined by

c (u_{1}, u_{2}, \dots, u_{m}; C) = Φ_{C} (ϕ_{C_{11}}^{- 1} (u_{1}), ϕ_{C_{22}}^{- 1} (u_{2}), \dots, ϕ_{C_{mm}}^{- 1} (u_{m}))

where

m

is the number of dimensions,

Φ_{C}

is the multivariate Normal density function with mean zero and covariance matrix

C

and

ϕ_{C_{ii}}^{- 1}

is the inverse of the univariate Normal density function with mean zero and variance

C_{ii}

Routine nag_rgsn_matrix_multi_normal (g05lyc) is used to generate a vector from a multivariate Normal distribution and function nag_prob_normal (g01eac) is used to convert each element of that vector into a uniformly distributed value between zero and one.

One of the initialization functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_copula_normal (g05rac).

4 References

Nelsen R B (1998) An Introduction to Copulas. Lecture Notes in Statistics 139 Springer

Sklar A (1973) Random variables: joint distribution functions and copulas Kybernetika 9 499–460

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: mode – IntegerInput

On entry: selects the operation to be performed.

$mode = 0$: Initialize and generate random numbers.
$mode = 1$: Initialize only (i.e., set up reference vector).
$mode = 2$: Generate random numbers using previously set up reference vector.

Constraint:

mode = 0

1

2

3: m – IntegerInput

On entry:

m

, the number of dimensions of the distribution.

Constraint:

m > 0

4: c[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array c must be at least

pdc \times m

The

(i, j)

th element of the matrix

C

is stored in

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

On entry: the covariance matrix of the distribution. Only the upper triangle need be set.

Constraint: c must be positive semidefinite to machine precision.

5: pdc – IntegerInput

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

Constraint:

pdc \geq m

6: n – IntegerInput

On entry:

n

, the number of random variates required.

Constraint:

n \geq 1

7: x[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

The

i

th element of the

j

th vector

X

is stored in

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

On exit: the array of pseudorandom multivariate Normal vectors.

8: pdx – IntegerInput

On entry: the stride used in the array x.

Constraints:

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

9: igen – IntegerInput

On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).

10: iseed[ $4$ ] – IntegerCommunication Array

On entry: contains values which define the current state of the selected generator.

On exit: contains updated values defining the new state of the selected generator.

11: r[lr] – doubleCommunication Array

On entry: if

mode = 2

, the reference vector as set up by nag_rngs_copula_normal (g05rac) in a previous call with

mode = 0

1

On exit: if

mode = 0

1

, the reference vector that can be used in subsequent calls to nag_rngs_copula_normal (g05rac) with

mode = 2

12: lr – IntegerInput

On entry: the dimension of the array r. If

mode = 2

, it must be the same as the value of lr specified in the prior call to nag_rngs_copula_normal (g05rac) with

mode = 0

1

Constraint:

lr > m \times (m + 1)

13: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, invalid value for igen. Ensure igen has not changed since random number generator was initialized.; On entry, $m = 〈value〉$ .
Constraint: $m > 0$ .; On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .; On entry, $mode = 〈value〉$ .
Constraint: $mode \geq 0$ and $mode \leq 2$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .; On entry, $pdc = 〈value〉$ .
Constraint: $pdc > 0$ .; On entry, $pdx = 〈value〉$ .
Constraint: $pdx > 0$ .
NE_INT_2: m is not the same as when r was set up in a previous call. Previous value $= 〈value〉$ and $m = 〈value〉$ .; On entry, $lr = 〈value〉$ , $(m \times m + 1) = 〈value〉$ .
Constraint: $lr \geq m \times m + 1$ .; On entry, $pdc = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdc \geq m$ .; On entry, $pdx = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdx \geq m$ .; On entry, $pdx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdx \geq n$ .
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.
NE_POS_DEF: The covariance matrix c is not positive semidefinite to machine precision.