g05rcf sets up a reference vector and generates an array of pseudorandom numbers from a Student's $t$ copula with $\nu $ degrees of freedom and covariance matrix $\frac{\nu}{\nu -2}C$.
where $m$ is the number of dimensions, ${T}_{\nu ,C}^{m}$ is the multivariate Student's $t$ density function with $\nu $ degrees of freedom, mean zero and covariance matrix $\frac{\nu}{\nu -2}C$ and ${t}_{\nu ,{C}_{\mathit{ii}}}^{-1}$ is the inverse of the univariate Student's $t$ density function with $\nu $ degrees of freedom, zero mean and variance $\frac{\nu}{\nu -2}{C}_{\mathit{ii}}$.
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.
4References
Nelsen R B (1998) An Introduction to Copulas. Lecture Notes in Statistics 139 Springer
Sklar A (1973) Random variables: joint distribution functions and copulas Kybernetika9 499–460
5Arguments
1: $\mathbf{mode}$ – IntegerInput
On entry: a code for selecting the operation to be performed by the routine.
${\mathbf{mode}}=0$
Set up reference vector only.
${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to g05rcf.
${\mathbf{mode}}=2$
Set up reference vector and generate variates.
Constraint:
${\mathbf{mode}}=0$, $1$ or $2$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of random variates required.
Constraint:
${\mathbf{n}}\ge 0$.
3: $\mathbf{df}$ – IntegerInput
On entry: $\nu $, the number of degrees of freedom of the distribution.
Constraint:
${\mathbf{df}}\ge 3$.
4: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of dimensions of the distribution.
Constraint:
${\mathbf{m}}>0$.
5: $\mathbf{c}({\mathbf{ldc}},{\mathbf{m}})$ – 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: $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which g05rcf is called.
Constraint:
${\mathbf{ldc}}\ge {\mathbf{m}}$.
7: $\mathbf{r}\left({\mathbf{lr}}\right)$ – Real (Kind=nag_wp) arrayCommunication Array
On entry: if ${\mathbf{mode}}=1$, the reference vector as set up by g05rcf in a previous call with ${\mathbf{mode}}=0$ or $2$.
On exit: if ${\mathbf{mode}}=0$ or $2$, the reference vector that can be used in subsequent calls to g05rcf with ${\mathbf{mode}}=1$.
8: $\mathbf{lr}$ – IntegerInput
On entry: the dimension of the array r as declared in the (sub)program from which g05rcf is called. If ${\mathbf{mode}}=1$, it must be the same as the value of lr specified in the prior call to g05rcf with ${\mathbf{mode}}=0$ or $2$.
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: $\mathbf{x}({\mathbf{ldx}},*)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
${\mathbf{m}}$.
On exit: the array of values from a multivariate Student's $t$ copula, with ${\mathbf{x}}(i,j)$ holding the $j$th dimension for the $i$th variate.
11: $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g05rcf is called.
Constraint:
${\mathbf{ldx}}\ge {\mathbf{n}}$.
12: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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, ${\mathbf{mode}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{df}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{df}}\ge 3$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}>0$.
${\mathbf{ifail}}=5$
On entry, the covariance matrix $C$ is not positive semidefinite to machine precision.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{ldc}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldc}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=7$
m is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=8$
On entry, lr is not large enough, ${\mathbf{lr}}=\u27e8\mathit{\text{value}}\u27e9$: minimum length required $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=9$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{ldx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
See Section 7 in g05ryf for an indication of the accuracy of the underlying multivariate Student's $t$-distribution.
8Parallelism and Performance
g05rcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05rcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
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
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 $C+E=L{L}^{\mathrm{T}}$, 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.
10Example
This example prints ten pseudorandom observations from a Student's $t$ copula with ten degrees of freedom and $C$ matrix
generated by g05rcf. All ten observations are generated by a single call to g05rcf with ${\mathbf{mode}}=2$. The random number generator is initialized by g05kff.