NAG Library Function Document
nag_rand_copula_students_t (g05rcc)
1
Purpose
nag_rand_copula_students_t (g05rcc) sets up a reference vector and generates an array of pseudorandom numbers from a Student's copula with degrees of freedom and covariance matrix .
2
Specification
#include <nag.h> |
#include <nagg05.h> |
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) |
|
3
Description
The Student's
copula,
, is defined by
where
is the number of dimensions,
is the multivariate Student's
density function with
degrees of freedom, mean zero and covariance matrix
and
is the inverse of the univariate Student's
density function with
degrees of freedom, zero mean and variance
.
nag_rand_matrix_multi_students_t (g05ryc) is used to generate a vector from a multivariate Student's
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).
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:
– 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
. 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:
or .
- 2:
– Nag_ModeRNGInput
-
On entry: a code for selecting the operation to be performed by the function.
- Set up reference vector only.
- Generate variates using reference vector set up in a prior call to nag_rand_copula_students_t (g05rcc).
- Set up reference vector and generate variates.
Constraint:
, or .
- 3:
– IntegerInput
-
On entry: , the number of random variates required.
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of degrees of freedom of the distribution.
Constraint:
.
- 5:
– IntegerInput
-
On entry: , the number of dimensions of the distribution.
Constraint:
.
- 6:
– const doubleInput
-
Note: the dimension,
dim, of the array
c
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: matrix which, along with
df, defines the covariance of the distribution. Only the upper triangle need be set.
Constraint:
must be positive semidefinite to machine precision.
- 7:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
c.
Constraint:
.
- 8:
– doubleCommunication Array
-
On entry: if , the reference vector as set up by nag_rand_copula_students_t (g05rcc) in a previous call with or .
On exit: if or , the reference vector that can be used in subsequent calls to nag_rand_copula_students_t (g05rcc) with .
- 9:
– IntegerInput
-
On entry: the dimension of the array
r. If
, it must be the same as the value of
lr specified in the prior call to
nag_rand_copula_students_t (g05rcc) with
or
.
Constraint:
.
- 10:
– IntegerCommunication Array
-
Note: the dimension,
, 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:
– doubleOutput
-
Note: the dimension,
dim, of the array
x
must be at least
- when ;
- when .
Where
appears in this document, it refers to the array element
- when ;
- when .
On exit: the array of values from a multivariate Student's copula, with holding the th dimension for the th variate.
- 12:
– IntegerInput
-
On entry: the stride used in the array
x.
Constraints:
- if ,
;
- if , .
- 13:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error 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 had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry,
lr is not large enough,
: minimum length required
.
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- 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 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
and
.
7
Accuracy
See
Section 7 in
nag_rand_matrix_multi_students_t (g05ryc) for an indication of the accuracy of the underlying multivariate Student's
-distribution.
8
Parallelism 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.
The time taken by nag_rand_copula_students_t (g05rcc) is of order .
It is recommended that the diagonal elements of should not differ too widely in order of magnitude. This may be achieved by scaling the variables if necessary. The actual matrix decomposed is , where is a diagonal matrix with small positive diagonal elements. This ensures that, even when is singular, or nearly singular, the Cholesky factor corresponds to a positive definite covariance matrix that agrees with within machine precision.
10
Example
This example prints ten pseudorandom observations from a Student's
copula with ten degrees of freedom and
matrix
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
. The random number generator is initialized by
nag_rand_init_repeatable (g05kfc).
10.1
Program Text
Program Text (g05rcce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (g05rcce.r)