The Joe/B5 copula (Joe, 2014, p. 170) is
$$\mathbf{C}_{\Theta}(u,v) = \mathbf{B5}(u,v) = 1 - \bigl((1-u)^\Theta + (1-v)^\Theta - (1-u)^\Theta (1-v)^\Theta\bigr)\mbox{,}$$
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.
JOcopB5(u, v, para=NULL, tau=NULL, ...)Value(s) for the copula are returned. Otherwise if tau is given, then the \(\Theta\) is computed and a list having
The parameter \(\Theta\), and
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.
Nonexceedance probability \(u\) in the \(X\) direction;
Nonexceedance probability \(v\) in the \(Y\) direction;
A vector (single element) of parameters---the \(\Theta\) parameter of the copula;
Optional Kendall Tau; and
Additional arguments to pass.
W.H. Asquith
Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.
M, P