FRcop: The Frank Copula

Description

The Frank copula (Joe, 2014, p. 165) is

$$\mathbf{C}_{\Theta}(u,v) = \mathbf{FR}(u,v) = -\frac{1}{\Theta}\mathrm{log}\biggl[\frac{1 - \mathrm{e}^{-\Theta} - \bigl(1 - \mathrm{e}^{-\Theta u}\bigr) \bigl(1 - \mathrm{e}^{-\Theta v}\bigr)}{1 - \mathrm{e}^{-\Theta}}\biggr]\mbox{,} $$

where $\Theta \in [-\infty, +\infty], \Theta \ne 0$. The copula, as $\Theta \rightarrow -\infty$ limits, to the countermonotonicity coupla ($\mathbf{W}(u,v)$; W), as $\Theta \rightarrow 0^{\pm}$ 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 numerical methods as $\tau_{\mathbf{C}}(\Theta) = 1 + 4\Theta^{-1}[D_1(\Theta) - 1]$ or from a Spearman Rho (rhoCOP) as $\rho_{\mathbf{C}}(\Theta) = 1 + 4\Theta^{-1}[D_2(\Theta) - D_1(\Theta)]$ for Debye function as $$ D_k(x, k) = k x^{-k} \int_0^x t^k \bigl(\mathrm{e}^{t} - 1\bigr)^{-1}\, \mathrm{d}t\mbox{.} $$

Usage

FRcop(u, v, para=NULL, rhotau=NULL, userhotau_chk=TRUE,
            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;
userhotau_chk: A logical to trigger computation of Kendall Tau for the given parameter and used as a secondary check on numerical limits of the copula implementation for the package; and
...: Additional arguments to pass.

Author

W.H. Asquith

References