NAG Library Routine Document

G05RCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G05RCF sets up a reference vector and generates an array of pseudorandom numbers from a Student's

t

copula with

ν

degrees of freedom and covariance matrix

\frac{ν}{ν - 2} C

2 Specification

SUBROUTINE G05RCF (

MODE, N, DF, M, C, LDC, R, LR, STATE, X, LDX, IFAIL)

INTEGER	MODE, N, DF, M, LDC, LR, STATE(*), LDX, IFAIL
REAL (KIND=nag_wp)	C(LDC,M), R(LR), X(LDX,M)

3 Description

The Student's

t

copula,

G

, is defined by

G (u_{1}, u_{2}, \dots, u_{m}; C) = T_{ν, C}^{m} (t_{ν, C_{11}}^{- 1} (u_{1}), t_{ν, C_{22}}^{- 1} (u_{2}), \dots, t_{ν, C_{mm}}^{- 1} (u_{m}))

where

m

is the number of dimensions,

T_{ν, C}^{m}

is the multivariate Student's

t

density function with

ν

degrees of freedom, mean zero and covariance matrix

\frac{ν}{ν - 2} C

and

t_{ν, C_{ii}}^{- 1}

is the inverse of the univariate Student's

t

density function with

ν

degrees of freedom, zero mean and variance

\frac{ν}{ν - 2} C_{ii}

G05RYF is used to generate a vector from a multivariate Student's

t

distribution and G01EBF is used to convert each element of that vector into a uniformly distributed value between zero and one.

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 G05RCF.

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 Parameters

1: MODE – INTEGERInput

On entry: a code for selecting the operation to be performed by the routine.

$MODE = 0$: Set up reference vector only.
$MODE = 1$: Generate variates using reference vector set up in a prior call to G05RCF.
$MODE = 2$: Set up reference vector and generate variates.

Constraint:

MODE = 0

1

2

2: N – INTEGERInput

On entry:

n

, the number of random variates required.

Constraint:

N \geq 0

3: DF – INTEGERInput

On entry:

ν

, the number of degrees of freedom of the distribution.

Constraint:

DF \geq 3

4: M – INTEGERInput

On entry:

m

, the number of dimensions of the distribution.

Constraint:

M > 0

5: C(LDC,M) – REAL (KIND=nag_wp) arrayInput

On entry: matrix which, along with DF, defines the covariance of the distribution. Only the upper triangle need be set.

Constraint:

C

must be positive semidefinite to machine precision.

6: LDC – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which G05RCF is called.

Constraint:

LDC \geq M

7: R(LR) – REAL (KIND=nag_wp) arrayCommunication Array

On entry: if

MODE = 1

, the reference vector as set up by G05RCF in a previous call with

MODE = 0

2

On exit: if

MODE = 0

2

, the reference vector that can be used in subsequent calls to G05RCF with

MODE = 1

8: LR – INTEGERInput

On entry: the dimension of the array R as declared in the (sub)program from which G05RCF is called. If

MODE = 1

, it must be the same as the value of LR specified in the prior call to G05RCF with

MODE = 0

2

Constraint:

LR \geq M \times (M + 1) + 2

9: STATE( $*$ ) – 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.

10: X(LDX,M) – REAL (KIND=nag_wp) arrayOutput

On exit: the array of values from a multivariate Student's

t

copula, with

X (i, j)

holding the

j

th dimension for the

i

th variate.

11: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G05RCF is called.

Constraint:

LDX \geq N

12: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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, $MODE \neq 0$ , $1$ or $2$ .

$IFAIL = 2$: On entry, $N < 0$ .

$IFAIL = 3$: On entry, $DF \leq 2$ .

$IFAIL = 4$: On entry, $M < 1$ .

$IFAIL = 5$: The covariance matrix $C$ is not positive semidefinite to machine precision.

$IFAIL = 6$: On entry, $LDC < M$ .

$IFAIL = 7$: The reference vector R has been corrupted or M has changed since R was set up in a previous call to G05RCF with $MODE = 0$ or $2$ .

$IFAIL = 8$: On entry, $LR \leq M \times (M + 1) + 1$ .

$IFAIL = 9$

On entry,

STATE vector was not initialized or has been corrupted.

$IFAIL = 11$: On entry, $LDX < N$ .

7 Accuracy

See Section 7 in G05RYF for an indication of the accuracy of the underlying multivariate Student's

t

-distribution.

8 Further Comments

The time taken by G05RCF is of order

n m^{3}

It is recommended that the diagonal elements of

C

should not differ too widely in order of magnitude. This may be achieved by scaling the variables if necessary. The actual matrix decomposed is

C + E = L L^{T}

, where

E

is a diagonal matrix with small positive diagonal elements. This ensures that, even when

C

is singular, or nearly singular, the Cholesky factor

L

corresponds to a positive definite covariance matrix that agrees with

C

within machine precision.

9 Example

This example prints ten pseudorandom observations from a Student's

t

copula with ten degrees of freedom and

C

matrix

[\begin{array}{r} 1.69 & 0.39 & - 1.86 & 0.07 \\ 0.39 & 98.01 & - 7.07 & - 0.71 \\ - 1.86 & - 7.07 & 11.56 & 0.03 \\ 0.07 & - 0.71 & 0.03 & 0.01 \end{array}],

generated by G05RCF. All ten observations are generated by a single call to G05RCF with

MODE = 2

. The random number generator is initialized by G05KFF.

NAG Library Routine DocumentG05RCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G05RCF