naginterfaces.library.rand.copula_​gumbel

naginterfaces.library.rand.copula_gumbel(n, m, theta, sorder, statecomm)

copula_gumbel generates pseudorandom uniform variates with joint distribution of a Gumbel–Hougaard Archimedean copula.

For full information please refer to the NAG Library document for g05rk

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g05/g05rkf.html

Parameters

nint

$n$, the number of pseudorandom uniform variates to generate.

mint

$m$, the number of dimensions.

thetafloat

$\theta$, the copula parameter.

sorderint

Determines the storage order of variates; the $(\text{i},\text{j})$th variate is stored in $x[\text{i}-1,\text{j}-1]$ if $sorder=1$, and $x[\text{j}-1,\text{i}-1]$ if $sorder=2$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

xfloat, ndarray, shape $(:,:)$

The pseudorandom uniform variates with joint distribution described by $C_{\theta }$, with $x[i-1,j-1]$ holding the $i$th value for the $j$th dimension if $sorder=1$ and the $j$th value for the $i$th dimension of $sorder=2$.

Raises

NagValueError

(errno $1$)

On entry, corrupt $statecomm$[‘state’] argument.

(errno $2$)

On entry, invalid $theta$: $theta=⟨value⟩$.

Constraint: $theta\ge 1.0$.

(errno $3$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $4$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 2$.

(errno $5$)

On entry, invalid $sorder$.

Constraint: $sorder=1$ or $2$.

Notes

Generates $n$ pseudorandom uniform $m$-variates whose joint distribution is the Gumbel–Hougaard Archimedean copula $C_{\theta }$, given by

$$\begin{array}{c}{C}_{\theta }=exp\{-[{(-ln\left(u_1\right))}^{\theta }+{(-ln\left(u_2\right))}^{\theta }+\cdots +{(-ln\left(u_m\right))}^{\theta }]\}\text{,}\{\begin{array}{cc}\theta \in (1,\infty )\text{,}& \\ u_j\in (0,1]\text{,}j=1,2,\dots m\text{;}& \end{array}\end{array}$$

with the special cases:

$C_1=u_1{u}_2\cdots u_m$, the product copula;

$C_{\infty }=min(u_1,u_2,\dots ,u_m)$, the Fréchet–Hoeffding upper bound.

The generation method uses mixture of powers.

One of the initialization functions init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to copula_gumbel.

References

Marshall, A W and Olkin, I, 1988, Families of multivariate distributions, Journal of the American Statistical Association (83), 403

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