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): $C_{ρ, ν} (u, v) = tEV (u, v; ρ, ν) = \exp (- (x + y) \times B (x / (x + y); ρ, ν)),$ where $x = - \log (u)$ , $y = - \log (v)$ , and letting $η = \sqrt{(ν + 1) / (1 - ρ^{2})}$ define $B (w; ρ, ν) = w \times T_{ν + 1} (η [(w / [1 - w])^{1 / ν} - ρ]) + (1 - w) \times T_{ν + 1} (η [([1 - w] / w)^{1 / ν} - ρ]),$ where $T_{ν + 1}$ is the cumulative distribution function of the univariate t-distribution with $ν - 1$ degrees of freedom. As $ν \to \infty$ , 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 $tEV (u, v; ρ, ν)$ 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 $ν$ and $ρ$ . 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 $ρ$ and $ν$ order; and
...: Additional arguments to pass.

Author

W.H. Asquith

References

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

Demarta, S., and McNeil, A.J., 2004, The t copula and related copulas: International Statistical Review, v. 33, no. 1, pp. 111--129, tools:::Rd_expr_doi("10.1111/j.1751-5823.2005.tb00254.x")

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