CLcop {copBasic}

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.

Author(s)

W.H. Asquith

References

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

See Also

M, P, W

Examples

# Lower tail dependency of Theta = pi --> 2^(-1/pi) = 0.8020089 (Joe, 2014, p. 168)
taildepCOP(cop=CLcop, para=pi)$lambdaL # 0.80201