TRIcop: The Triangular Copula

In copBasic: General Bivariate Copula Theory and Many Utility Functions

The Triangular Copula

Description

The Triangular copula (Nelsen, 2006, exam. 3.3, pp. 59–60; exer. 3.7, pp. 64–65) is

\mathbf{C}_{\Theta}(u,v) = \mathbf{TRI}(u,v)

for \Theta \in (0,-1). The copula, as \Theta \rightarrow 0 limits, to the countermonotonicity coupla (\mathbf{W}(u,v); W), as \Theta \rightarrow 1 limits to the comonotonicity copula (\mathbf{M}(u,v); M). The parameter \Theta is readily computed from a Spearman Rho (rhoCOP) or Kendall Tau (tauCOP) by \rho_\mathbf{C} = \tau_\mathbf{C} = 2\Theta-1. However in general for finite samples, the sample versions of Spearman Rho and Kendall Tau will not be equal and different \Theta wil result.

Usage

TRIcop(u, v, para=NULL, rhotau=NULL, ...)

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 Spearman Rho or Kendall Tau; and
...	Additional arguments to pass.

Value

Value(s) for the copula are returned. Otherwise if rhotau is given, then the \Theta is computed and a list having

para	The parameter \Theta, and
rhotau	Spearman Rho or Kendall Tau.

and if para=NULL and rhotau=NULL, then the values within u and v are used to compute Spearman Rho or Kendall Tau and then compute the parameter, and these are returned in the aforementioned list.

Author(s)

W.H. Asquith

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

See Also

M, W

Examples

# Concerning Nelsen (2006, exer. 5.6, p. 171)
rhoCOP(cop=TRIcop, para=0.25) # -0.4999916
tauCOP(cop=TRIcop, para=0.25) # -0.5

## Not run: 
  for(i in seq_len(4)) {
    para <- list(cop=TRIcop, para=0.25, reflect=i)
    uv <- simCOP(100, cop=COP, para=para, seed=1, col=i, ploton=! as.logical(i-1))
  } # 
## End(Not run)

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