void	nag_rand_bivariate_copula_plackett (Nag_OrderType order, Integer state[], double theta, Integer n, double x[], Integer pdx, Integer sdx, NagError *fail)

3 Description

Generates pseudorandom uniform bivariates

\{u_{1}, u_{2}\} \in {[0, 1]}^{2}

whose joint distribution is the Plackett copula

C_{θ}

with parameter

θ

, given by

C_{θ} = \frac{[1 + (θ - 1) (u_{1} + u_{2})] - \sqrt{{[1 + (θ - 1) (u_{1} + u_{2})]}^{2} - 4 u_{1} u_{2} θ (θ - 1)}}{2 (θ - 1)}, θ \in (0, \infty) ∖ \{1\}

with the special cases:

$C_{0} = \max (u_{1} + u_{2} - 1, 0)$ , the Fréchet–Hoeffding lower bound;
$C_{1} = u_{1} u_{2}$ , the product copula;
$C_{\infty} = \min (u_{1}, u_{2})$ , the Fréchet–Hoeffding upper bound.

The generation method uses conditional sampling.

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_bivariate_copula_plackett (g05rgc).

4 References

Nelsen R B (2006) An Introduction to Copulas (2nd Edition) Springer Series in Statistics

5 Arguments

1: $order$ – 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

order = Nag_RowMajor

. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $state [\dim]$ – IntegerCommunication Array

Note: the dimension,

\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.

3: $theta$ – doubleInput

On entry:

θ

, the copula parameter.

Constraint:

theta \geq 0.0

4: $n$ – IntegerInput

On entry:

n

, the number of bivariates to generate.

Constraint:

n \geq 0

5: $x [pdx \times sdx]$ – doubleOutput

Note: where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

n

bivariate uniforms with joint distribution described by

C_{θ}

, with

X (i, j)

holding the

i

th value for the

j

th dimension if

order = Nag_ColMajor

and the

j

th value for the

i

th dimension if

order = Nag_RowMajor

6: $pdx$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq 2$ .

7: $sdx$ – IntegerInput

On entry: the secondary dimension of X.

Constraints:

if $order = Nag_ColMajor$ , $sdx \geq 2$ ;
if $order = Nag_RowMajor$ , $sdx \geq n$ .

8: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 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 $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, pdx must be at least $〈value〉$ : $pdx = 〈value〉$ .

On entry, sdx must be at least $〈value〉$ : $sdx = 〈value〉$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_STATE: On entry, corrupt state argument.
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_REAL: On entry, invalid theta: $theta = 〈value〉$ .
Constraint: $theta \geq 0.0$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

nag_rand_bivariate_copula_plackett (g05rgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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.

9 Further Comments

In practice, the need for numerical stability restricts the range of

θ

such that:

if $θ < ε_{s}$ , the function returns pseudorandom uniform variates with $C_{0}$ joint distribution;
if $|θ - 1| < ε$ , the function returns pseudorandom uniform variates with $C_{1}$ joint distribution;
if $θ > ε_{s}^{- 1 / 2}$ , the function returns pseudorandom uniform variates with $C_{\infty}$ joint distribution;