tEVcop: The t-EV (Extreme Value) Copula

Description

The t-EV copula (Joe, 2014, p. 189) is a limiting form of the t-copula (multivariate t-distribution): $$ \mathbf{C}_{\rho,\nu}(u,v) = \mathbf{tEV}(u,v; \rho, \nu) = \mathrm{exp}\bigl(-(x+y) \times B(x/(x+y); \rho, \nu)\bigr)\mbox{,} $$ where $x = -\log(u)$, $y = -\log(v)$, and letting $\eta = \sqrt{(\nu+1)/(1-\rho^2)}$ define $$ B(w; \rho, \nu) = wT_{\nu+1}\bigl(\eta[(w/[1-w])^{1/\nu}-\rho]\bigr) + (1-w)T_{\nu+1}\bigl(\eta[([1-w]/w)^{1/\nu}-\rho]\bigr)\mbox{,} $$ where $T_{\nu+1}$ is the cumulative distribution function of the univariate t-distribution with $\nu-1$ degrees of freedom. As $\nu \rightarrow \infty$, the copula weakly converges to the Hüsler--Reiss copula (HRcop) because the t-distribution converges to the normal (see Examples for a study of this copula).

The $\mathbf{tEV}(u,v; \rho, \nu)$ copula is a two-parameter option when working with extreme-value copula. There is a caveat though. Demarta and McNeil (2004) conclude that “the parameter of the Gumbel [GHcop] or Galambos [GLcop] A-functions [the Pickend dependence function and B-function by association] can always be chosen so that the curve is extremely close to that of the t-EV A-function for any values of $\nu$ and $\rho$. The implication is that in all situations where the t-EV copula might be deemed an appropriate model then the practitioner can work instead with the simpler Gumbel or Galambos copulas.”

Usage

tEVcop(u, v, para=NULL, ...)

Value

Value(s) for the copula are returned.

Arguments

u: Nonexceedance probability $u$ in the $X$ direction;
v: Nonexceedance probability $v$ in the $Y$ direction;
para: A vector (two element) of parameters in $\rho$ and $\nu$ order; and
...: Additional arguments to pass.

Author

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.