JOcopB5: The Joe/B5 Copula (B5)

In copBasic: General Bivariate Copula Theory and Many Utility Functions

The Joe/B5 Copula (B5)

Description

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

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

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

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

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

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

where \psi' is the trigamma() function.

Usage

JOcopB5(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

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)

## Not run: 
   # embeddment of parameters for permutation asymmetry and rotation
   para <- list(cop=JOcopB5, para=30, reflect=3, breve=0.5)
   UV <- simCOP(cop=COP, para=list(cop=breveCOP, para=para), n=1000) #
## End(Not run)

copBasic documentation built on June 28, 2025, 1:08 a.m.