breveCOP: Add Asymmetry to a Copula

Description

Adding permutation asymmetry (Chang and Joe, 2020, p. 1596) (isCOP.permsym) is simple for a bivariate copula family. Let $\mathbf{C}$ be a copula with respective vectors of parameters $\Theta_\mathbf{C}$, then the permutation asymmetry is added through an asymmetry parameter $\beta \in (-1, +1)$ by

$$\breve{\mathbf{C}}_{\beta;\Theta}(u,v) = v^{-\beta}\cdot\mathbf{C}\bigl(u, v^{(1+\beta)};\Theta\bigr)\mbox{, and}$$

for $0 \le \beta \le +1$ by

$$\breve{\mathbf{C}}_{\beta;\Theta}(u,v) = u^{+\beta}\cdot\mathbf{C}\bigl(u^{(1-\beta)}, v;\Theta\bigr)\mbox{.}$$ The parameter $\beta$ clashes in name and symbology with a parameter used by functions composite1COP, composite2COP, and composite3COP. As a result, support for alternative naming is provided for compatibility. Lastly, an example under W demonstrates the span of the Kendall function (kfuncCOP) based on M and W having permutation asymmetry inserted by breveCOP(). The demonstration shows that the Kendall function is a distribution unto itself with its own nonexceedance probabilities.

Usage

breveCOP(u,v, para, breve=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 special parameter list (see Note);
breve: An alternative way from para to set the $\beta$ for this function; and
...: Additional arguments to pass to the copula.

Author

W.H. Asquith

References