NORMcop: The Normal (Gaussian) Copula

Description

The Normal copula (Gaussian copula) (Salvadori et al., 2007, pp. 255--256) is

$$\mathbf{C}_{\Theta}(u,v) = \mathbf{NORM}(u,v; \Theta) = \int_{-\infty}^{\Phi^{(-1)}(u)}\!\!\!\!\int_{-\infty}^{\Phi^{(-1)}(v)}\!\!\!\!\!\!\!\! \frac{1}{2\pi\sqrt{1-\Theta^2}} \mathrm{exp}\biggl(-\frac{s^2 - 2\Theta s t + t^2}{ 2(1-\Theta^2) } \biggr)\,\mathrm{d}s\,\mathrm{d}t\mbox{,}$$

where $\Theta \in [-1,1]$ and $\Phi^{(-1)}(x)$ is the quantile function of the univariate normal distribution. The copula, as $\Theta \rightarrow -1^{+}$, limits to the countermonotonicity copula ($\mathbf{W}(u,v)$; W), as $\Theta \rightarrow 0$ limits, to the independence coupla ($\mathbf{P}(u,v)$; P), and as $\Theta \rightarrow 1^{-}$, limits to the comonotonicity copula ($\mathbf{M}(u,v)$; M). The copula has lower-tail and upper-tail dependency parameters equal to zero, but such is not true for the closely related t-Student copula (Tcop). The Spearman Rho (rhoCOP) is $\rho_\mathbf{C} = (6/\pi)\cdot\mathrm{asin}(\Theta/2)$ and Kendall Tau (tauCOP) is $\tau_\mathbf{C} = (2/\pi)\cdot\mathrm{asin}(\Theta)$. The parameter $\Theta$ is readily computed by $\Theta = 2\cdot\mathrm{sin}(\pi\cdot\rho_\mathbf{C}/6)$ or by $\Theta = \mathrm{sin}(\pi\cdot\tau_\mathbf{C}/2)$.

Usage

NORMcop(u, v, para=NULL, rho=NULL, tau=NULL, fit=c("rho", "tau"), ...)

Value

Value(s) for the copula are returned. Otherwise if either rho or tau is given, then the $\Theta$ is computed and a list having

para: The parameter $\Theta$;
rho: Spearman Rho if the rho is given; and
tau: Kendall Tau if the tau is given but also if both rho and tau are NULL as mentioned next.

and if para=NULL and rho and tau=NULL, then the values within u and v are used to compute Spearman Rho (fit="rho") or Kendall Tau (fit="tau") and then compute the parameter, and this is returned in the aforementioned list.

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;
rho: Optional Spearman Rho from which the parameter will be estimated and presence of rho trumps tau;
tau: Optional Kendall Tau from which the parameter will be estimated;
fit: If para, rho, and tau are all NULL, then the u and v represent the sample. The measure of association by the fit declaration will be computed and the parameter estimated subsequently. The fit has no other utility than to trigger which measure of association is computed internally by the cor function in R; and
...: Additional arguments to pass.

Author

W.H. Asquith

References