The Hairpin copula (Durante et al., 2014) is
$$\mathbf{C}_{\Theta}(u,v) = \mathbf{HAIRPIN}(u,v) = \mathrm{min}\bigl[u, v, \mathrm{mean}\bigl(f(u,v) \bigr) \bigr]\mbox{,}$$
as vectorized for the pairs \((u_i, v_i)\) for \(i \in 1,2,\cdots\), and \(f(x) = \alpha x^\phi\), \(1 \le \alpha \le \infty\), and \(1 \le \phi \le 2\). The copula, as \(\alpha \rightarrow \infty\) or \(\phi \rightarrow 1\) limits, to the comonotonicity coupla (\(\mathbf{M}(u,v)\); M).
HAIRPINcop(u, v, para=c(1, 2, 1), ...)Value(s) for the copula are returned.
Nonexceedance probability \(u\) in the \(X\) direction;
Nonexceedance probability \(v\) in the \(Y\) direction;
A vector (three element) of two parameters and the reflection: \(\alpha\) and \(\phi\) of the coupla then the reflection number \(n \in 1, 2, 3, 4\) following the reflection pattern of COP; and
Additional arguments to pass.
W.H. Asquith
Durante, F., Fernández-Sánchez, J., and Trutschnig, W., 2014, Multivariate copulas with hairpin support: Journal of Multivariate Analysis, v. 130, pp. 323--334, tools:::Rd_expr_doi("10.1016/j.jmva.2014.06.009").
M