Tcop: The t-Student Copula

Description

The t-Student copula (Salvadori et al., 2007, pp. 255--256) is

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

where $\Theta \in [-1,1]$, $\nu \ge 1$ (integer), and $t_\nu^{(-1)}(x)$ is the quantile function of the univariate t-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 ($\mathbf{P}(u,v)$ if $\nu$ becomes large; P), and as $\Theta \rightarrow 1^{-}$, limits to the comonotonicity copula ($\mathbf{M}(u,v)$; M). The copula has lower-tail dependency and upper-tail dependency parameters that are nonzero if $\Theta > 0$ and both parameters equal to

$$\lambda_{L|U} = 2t_{\nu+1}\biggl(-\frac{\sqrt{\nu+1}\sqrt{1-\Theta}}{\sqrt{1+\Theta}}\biggr)\mathrm{,}$$

which tend to zero as $\nu \rightarrow \infty$, which are those for the Normal copula (NORMcop). The Spearman Rho (rhoCOP) is $\rho_\mathbf{C} = (6/\pi)\cdot\mathrm{asin}(\Theta/2)$ and Kendall Tau (tauCOP) is $\tau_\mathbf{C} = (2/\pi)\cdot\mathrm{asin}(\Theta)$ and are the same as for NORMcop. 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

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

Value

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

para: The parameters $\Theta$ and $\nu$ (if computable, refer to message) and note that $\nu$ will be silently cast as an integer after rounding to zero digits internally for mvtnorm::pmvt();
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.
lambdaUL: Computed tail dependency (if computable);
message: Helpful message;
lambdaUL_if_nu_were_eqone: Computed tail dependency if $\nu$ were to equal 1;

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 $\Theta$ and then attempt to compute the $\nu$ from the taildep, and these are 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 (two element) of parameters---$\Theta$ and $\nu$ parameters of the copula and note that $\nu$ will be silently cast as an integer after rounding to zero digits internally for mvtnorm::pmvt();
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;
taildep: Optional lower/upper tail dependency coefficient to try to fit the parameters for the give rho or tau;
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