JOcopB5 {copBasic}R Documentation

The Joe/B5 Copula (B5)

Description

The Joe/B5 copula (Joe, 2014, p. 170) is

CΘ(u,v)=B5(u,v)=1((1u)Θ+(1v)Θ(1u)Θ(1v)Θ)\mbox,\mathbf{C}_{\Theta}(u,v) = \mathbf{B5}(u,v) = 1 - \bigl((1-u)^\Theta + (1-v)^\Theta - (1-u)^\Theta (1-v)^\Theta\bigr)\mbox{,}

where Θ[1,)\Theta \in [1,\infty). The copula as Θ\Theta \rightarrow \infty limits to the comonotonicity coupla (M(u,v)\mathbf{M}(u,v) and M), as Θ1+\Theta \rightarrow 1^{+} limits to independence copula (Π(u,v)\mathbf{\Pi}(u,v); P). Finally, the parameter Θ\Theta is readily computed from a Kendall Tau (tauCOP) by

τC=1+22Θ(ψ(2)ψ(1+2/Θ))\mbox,\tau_\mathbf{C} = 1 + \frac{2}{2-\Theta}\bigl(\psi(2) - \psi(1 + 2/\Theta)\bigr)\mbox{,}

where ψ\psi is the digamma() function and as Θ2\Theta \rightarrow 2 then

τC(Θ2)=1ψ(2)\tau_\mathbf{C}(\Theta \rightarrow 2) = 1 - \psi'(2)

where ψ\psi' is the trigamma() function.

Usage

JOcopB5(u, v, para=NULL, tau=NULL, ...)

Arguments

u

Nonexceedance probability uu in the XX direction;

v

Nonexceedance probability vv in the YY direction;

para

A vector (single element) of parameters—the Θ\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 Θ\Theta is computed and a list having

para

The parameter Θ\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

Examples

# Upper tail dependency of Theta = pi --> 2 - 2^(1/pi) = 0.753131 (Joe, 2014, p. 171).
taildepCOP(cop=JOcopB5, para=pi)$lambdaU # 0.75313

# Blomqvist Beta of Theta = pi (Joe, 2014, p. 171).
blomCOP(cop=JOcopB5, para=pi)        # 0.5521328
3 - 4*(2*(1/2)^pi - (1/4)^pi)^(1/pi) # 0.5521328

## Not run: 
# A test near the limiting Theta for trigamma()
UV <- simCOP(cop=JOcopB5, para=2, n=10000)
para <- JOcopB5(UV[,1], UV[,2])$para
message("Tau difference ", round(2-para, digits=2), " is small.") #
## End(Not run)

[Package copBasic version 2.2.4 Index]