HRcop: The Hüsler-Reiss Extreme Value Copula

In copBasic: General Bivariate Copula Theory and Many Utility Functions

The Hüsler–Reiss Extreme Value Copula

Description

The Hüsler–Reiss copula (Joe, 2014, p. 176) is

\mathbf{C}_{\Theta}(u,v) = \mathbf{HR}(u,v) = \mathrm{exp}\bigr[-x \Phi(X) - y\Phi(Y)\bigr]\mbox{,}

where \Theta \ge 0, x = - \log(u), y = - \log(v), \Phi(.) is the cumulative distribution function of the standard normal distribution, X and Y are defined as:

X = \frac{1}{\Theta} + \frac{\Theta}{2} \log\biggl(\frac{x}{y}\biggr)\mbox{\ and\ } Y = \frac{1}{\Theta} + \frac{\Theta}{2} \log\biggl(\frac{y}{x}\biggr)\mbox{.}

As \Theta \rightarrow 0^{+}, the copula limits to independence (\mathbf{\Pi}; P). The \mathbf{HR} copula is a bivariate extreme value copula (BEV), and the parameter \Theta requires numerical methods. Because there is “hardly any practical difference among [these] bivariate exchangeable parametric families” distinguishing between these (GHcop, GLcop, and HRcop) “requires extremely large sample sizes” (Hofert et al., 2016, p. 117). This observation is strongly supported by the L-comoment ratio diagrams explored in LCOMDIA_ManyCops.

Usage

HRcop(u, v, para=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; and
...	Additional arguments to pass.

Value

Value(s) for the copula are returned.

Author(s)

W.H. Asquith

References

Hofert, M., Kojadinovic, I., Mächler, M., and Yan, J., 2018, Elements of copula modeling with R: Dordrecht, Netherlands, Springer.

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

See Also

P, GHcop, GLcop, tEVcop

Examples

# Parameter Theta = pi recovery through the Blomqvist Beta (Joe, 2014, p. 176)
qnorm(1 - log( 1 + blomCOP(cop=HRcop, para=pi) ) / ( 2 * log(2) ) )^(-1)

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