copBasic (version 2.1.5)

CLcop: The Clayton Copula

Description

The Clayton copula (Joe, 2014, p. 168) is $$\mathbf{C}_{\Theta}(u,v) = \mathbf{CL}(u,v) = \mathrm{max}\bigl[(u^{-\Theta}+v^{-\Theta}-1; 0)\bigr]^{-1/\Theta}\mbox{,}$$ where \(\Theta \in [-1,\infty), \Theta \ne 0\). The copula, as \(\Theta \rightarrow -1^{+}\) limits, to the countermonotonicity coupla (\(\mathbf{W}(u,v)\); W), as \(\Theta \rightarrow 0\) limits to the independence copula (\(\mathbf{\Pi}(u,v)\); P), and as \(\Theta \rightarrow \infty\), limits to the comonotonicity copula (\(\mathbf{M}(u,v)\); M). The parameter \(\Theta\) is readily computed from a Kendall Tau (tauCOP) by \(\tau_\mathbf{C} = \Theta/(\Theta+2)\).

Usage

CLcop(u, v, para=NULL, tau=NULL, ...)

Arguments

u

Nonexceedance probability \(u\) in the \(X\) direction;

v

Nonexceedance probability \(v\) in the \(Y\) direction;

para

A vector (single element) of parameters---the \(\Theta\) parameter of the copula;

tau

Optional Kendall Tau; and

...

Additional arguments to pass.

Value

Value(s) for the copula are returned. Otherwise if tau is given, then the \(\Theta\) is computed and a list having

para

The parameter \(\Theta\), and

tau

Kendall tau.

and if para=NULL and tau=NULL, then the values within u and v are used to compute Kendall Tau and then compute the parameter, and these are returned in the aforementioned list.

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

See Also

M, P, W

Examples

# NOT RUN {
# Lower tail dependency of Theta = pi --> 2^(-1/pi) = 0.8020089 (Joe, 2014, p. 168)
taildepCOP(cop=CLcop, para=pi)$lambdaL # 0.80201
# }