JOMAcop: The Joe--Ma Copula

Description

The Joe--Ma copula (Joe, 2014, p. 177), crediting Joe and Ma (2000), is

$$\mathbf{C}_{\Theta}(u,v) = \mathbf{JOMA}(u,v) = 1 - F_{\Gamma}\biggl(\biggl\{ \bigl[F_\Gamma^{(-1)}(1-u; \Theta)\bigr]^\Theta + \bigl[F_\Gamma^{(-1)}(1-v; \Theta)\bigr]^\Theta \biggr\}^{(1/\Theta)}; \Theta\biggr)\mbox{,} $$

where $F_\Gamma(x;\Theta)$ is the cumulative distribution function of the Gamma distribution, $F_\Gamma^{(-1)}(f;\Theta)$ is the quantile function of the Gamma distribution, and $\Theta \in (-1, +1)$, which is a different parameter range than shown in Joe (2014, p. 177) because of numerical difficulties with the Gamma distribution and functional performance in context of copBasic design. The copula limits, as $\Theta \rightarrow -1$, to the countermonotonicity coupla ($\mathbf{W}(u,v)$; W), as $\Theta \rightarrow 1$, to the comonotonicity copula ($\mathbf{M}(u,v)$; M), and as $\Theta \rightarrow 0^{\pm}$, to the independence copula ($\mathbf{\Pi}(u,v)$; P), and as $\Theta \rightarrow +\infty$. The parameter $\Theta$ is readily computed from Kendall Tau (tauCOP) or Spearman Rho (rhoCOP) by root-solving methods. Because the formulation here uses a differing parameter range than shown in Joe (2014), integral formulations for Kendall Tau are not shown in this documentation.

Usage

JOMAcop(u, v, para=NULL, rhotau=NULL,
              cortype=c("kendall", "spearman", "tau", "rho"), ...)

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. Or if rho is given, then the $\Theta$ is computed and a similar list is returned having similar structure but with Spearman Rho instead.

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;
rhotau: Optional Kendall Tau or Spearman Rho and parameter para is returned depending on the setting of cortype. The u and v can be used for estimation of the parameter as computed through the setting of cortype;
cortype: A character string controlling, if the parameter is not given, to use a Kendall Tau or Spearman Rho for estimation of the parameter. The name of this argument is reflective of an internal call to stats::cor() to the correlation (association) setting for Kendall Tau or Spearman Rho; and
...: Additional arguments to pass.

Author

W.H. Asquith

References

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