# NAG Library Function Document

## 1Purpose

nag_rand_copula_students_t (g05rcc) 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 $\frac{\nu }{\nu -2}C$.

## 2Specification

 #include #include
 void nag_rand_copula_students_t (Nag_OrderType order, Nag_ModeRNG mode, Integer n, Integer df, Integer m, const double c[], Integer pdc, double r[], Integer lr, Integer state[], double x[], Integer pdx, NagError *fail)

## 3Description

The Student's $t$ copula, $G$, is defined by
 $G u1 , u2 ,…, um ; C = T ν,C m t ν,C11 -1 u1 , t ν,C22 -1 u2 ,…, t ν,Cmm -1 um$
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 $\frac{\nu }{\nu -2}{C}_{\mathit{ii}}$.
nag_rand_matrix_multi_students_t (g05ryc) is used to generate a vector from a multivariate Student's $t$ distribution and 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_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_students_t (g05rcc).

## 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 Kybernetika 9 499–460

## 5Arguments

1:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. 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: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{mode}$Nag_ModeRNGInput
On entry: a code for selecting the operation to be performed by the function.
${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$
Set up reference vector only.
${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$
Generate variates using reference vector set up in a prior call to nag_rand_copula_students_t (g05rcc).
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
Set up reference vector and generate variates.
Constraint: ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$, $\mathrm{Nag_GenerateFromReference}$ or $\mathrm{Nag_InitializeAndGenerate}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of random variates required.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{df}$IntegerInput
On entry: $\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df}}\ge 3$.
5:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of dimensions of the distribution.
Constraint: ${\mathbf{m}}>0$.
6:    $\mathbf{c}\left[\mathit{dim}\right]$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.
Constraint: $C$ must be positive semidefinite to machine precision.
7:    $\mathbf{pdc}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{m}}$.
8:    $\mathbf{r}\left[{\mathbf{lr}}\right]$doubleCommunication Array
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector as set up by nag_rand_copula_students_t (g05rcc) in a previous call with ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$.
On exit: if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$, the reference vector that can be used in subsequent calls to nag_rand_copula_students_t (g05rcc) with ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$.
9:    $\mathbf{lr}$IntegerInput
On entry: the dimension of the array r. If ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, it must be the same as the value of lr specified in the prior call to nag_rand_copula_students_t (g05rcc) with ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$.
Constraint: ${\mathbf{lr}}\ge {\mathbf{m}}×\left({\mathbf{m}}+1\right)+2$.
10:  $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, 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.
11:  $\mathbf{x}\left[\mathit{dim}\right]$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Where ${\mathbf{X}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ 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 values from a multivariate Student's $t$ copula, with ${\mathbf{X}}\left(i,j\right)$ holding the $j$th dimension for the $i$th variate.
12:  $\mathbf{pdx}$IntegerInput
On entry: the stride used in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{m}}$.
13:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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 $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 3$.
On entry, lr is not large enough, ${\mathbf{lr}}=〈\mathit{\text{value}}〉$: minimum length required $\text{}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}>0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
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_POS_DEF
On entry, the covariance matrix $C$ is not positive semidefinite to machine precision.
NE_PREV_CALL
m is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.

## 7Accuracy

See Section 7 in nag_rand_matrix_multi_students_t (g05ryc) for an indication of the accuracy of the underlying multivariate Student's $t$-distribution.

## 8Parallelism and Performance

nag_rand_copula_students_t (g05rcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_rand_copula_students_t (g05rcc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

The time taken by nag_rand_copula_students_t (g05rcc) 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
 $1.69 0.39 -1.86 0.07 0.39 98.01 -7.07 -0.71 -1.86 -7.07 11.56 0.03 0.07 -0.71 0.03 0.01 ,$
generated by nag_rand_copula_students_t (g05rcc). All ten observations are generated by a single call to nag_rand_copula_students_t (g05rcc) with ${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$. The random number generator is initialized by nag_rand_init_repeatable (g05kfc).

### 10.1Program Text

Program Text (g05rcce.c)

None.

### 10.3Program Results

Program Results (g05rcce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017