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” distiguishing 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, ...)
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 (single element) of parameters---the \(\Theta\) parameter of the copula; and
...
Additional arguments to pass.
Author
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.