Tcop: The t-Student Copula

In copBasic: General Bivariate Copula Theory and Many Utility Functions

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"), ...)

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.

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.

Note

A mimic of the wrappers on mvtnorm::pmvt() from the copula package (v1.1-6) are made within Tcop. The rest of the implementation is unique to copBasic. An implementation of a function in the copBasic style that would natively interconnect to copula package version of the copula could be as follows:

  "Tcop" <-         # pCoupla() from package copula is analogous to COP()
  function(u,v, para=NULL) {
    if(length(u) == 1) u <- rep(u, length(v)) # see asCOP() for reasoning of
    if(length(v) == 1) v <- rep(v, length(u)) # this "vectorization" hack
    para <- copula::tCopula(c(para[1]), dim=2, df=as.integer(para[2]))
    return( copula::pCopula(matrix(c(u,v), ncol=2), para) )
  }

Author(s)

W.H. Asquith

References

Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature—An approach using copulas: Springer, 289 p.

See Also

NORMcop

Examples

## Not run: 
  para <- Tcop(para=NULL, rho=0.9, taildep=0.3) # 0.907981 22.000000
  UV <- simCOP(1000, cop=Tcop, para=para$para)  # see the tail dependency 
## End(Not run)

copBasic documentation built on July 23, 2026, 1:07 a.m.