g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rngs_copula_students_t (g05rbc)

## 1  Purpose

nag_rngs_copula_students_t (g05rbc) 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 $\frac{\nu }{\nu -2}C$.

## 2  Specification

 #include #include
 void nag_rngs_copula_students_t (Nag_OrderType order, Integer mode, Integer df, 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 Student's $t$ copula, $c$, is defined by
 $c u1 , u2 ,…, um ; C = T ν,C m t ν,C11 -1 u1 , t ν,C22 -1 u2 ,…, t ν,Cmm -1 um$
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 $\frac{\nu }{\nu -2}{C}_{\mathit{ii}}$.
Routine nag_rngs_matrix_multi_students_t (g05lxc) is used to generate a vector from a multivariate Student's $t$ distribution and function nag_prob_students_t (g01ebc) 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_students_t (g05rbc).

## 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:     orderNag_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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     modeIntegerInput
On entry: selects the operation to be performed.
${\mathbf{mode}}=0$
Initialize and generate random numbers.
${\mathbf{mode}}=1$
Initialize only (i.e., set up reference vector).
${\mathbf{mode}}=2$
Generate random numbers using previously set up reference vector.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
3:     dfIntegerInput
On entry: $\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df}}\ge 3$.
4:     mIntegerInput
On entry: $m$, the number of dimensions of the distribution.
Constraint: ${\mathbf{m}}>0$.
5:     c[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array c must be at least ${\mathbf{pdc}}×{\mathbf{m}}$.
The $\left(i,j\right)$th element of the matrix $C$ is stored in
• ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{c}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: matrix which, along with df, defines the covariance of the distribution. Only the upper triangle need be set.
6:     pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{m}}$.
7:     nIntegerInput
On entry: $n$, the number of random variates required.
Constraint: ${\mathbf{n}}\ge 1$.
8:     x[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{m}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $i$th element of the $j$th vector $X$ is stored in
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the array of pseudorandom multivariate Student's $t$ vectors.
9:     pdxIntegerInput
On entry: the stride used in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{m}}$.
10:   igenIntegerInput
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).
11:   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.
12:   r[lr]doubleCommunication Array
On entry: if ${\mathbf{mode}}=2$, the reference vector as set up by nag_rngs_copula_students_t (g05rbc) in a previous call with ${\mathbf{mode}}=0$ or $1$.
On exit: if ${\mathbf{mode}}=0$ or $1$, the reference vector that can be used in subsequent calls to nag_rngs_copula_students_t (g05rbc) with ${\mathbf{mode}}=2$.
13:   lrIntegerInput
On entry: the dimension of the array r. If ${\mathbf{mode}}=2$, it must be the same as the value of lr specified in the prior call to nag_rngs_copula_students_t (g05rbc) with ${\mathbf{mode}}=0$ or $1$.
Constraint: ${\mathbf{lr}}>{\mathbf{m}}×\left({\mathbf{m}}+1\right)+1$.
14:   failNagError *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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}>2$.
On entry, invalid value for igen. Ensure igen has not changed since random number generator was initialized.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}>0$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}\ge 0$ and ${\mathbf{mode}}\le 2$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}>0$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}>0$.
NE_INT_2
m is not the same as when r was set up in a previous call. Previous value $=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{lr}}=〈\mathit{\text{value}}〉$, $\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}+1\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lr}}\ge {\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}+1$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
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.

## 7  Accuracy

See Section 7 in nag_rngs_matrix_multi_students_t (g05lxc) for an indication of the accuracy of the underlying multivariate $t$-distribution.

None.

## 9  Example

This example prints ten pseudorandom observations from a Student's $t$ copula with ten degrees of freedom and c matrix
 $1.69 0.39 -1.86 0.07 0.39 98.01 -7.07 -0.71 -1.86 -7.07 11.56 0.03 0.07 -0.71 0.03 0.01 ,$
generated by nag_rngs_copula_students_t (g05rbc). All ten observations are generated by a single call to nag_rngs_copula_students_t (g05rbc) with ${\mathbf{mode}}=0$. The random number generator is initialized by nag_rngs_init_repeatable (g05kbc).

### 9.1  Program Text

Program Text (g05rbce.c)

None.

### 9.3  Program Results

Program Results (g05rbce.r)