GLcop: The Galambos Extreme Value Copula (with Gamma Power Mixture [Joe/BB4] and Lower Extreme Value Limit)

Description

The Galambos copula (Joe, 2014, p. 174) is $$\mathbf{C}_{\Theta}(u,v) = \mathbf{GL}(u,v) = uv\,\mathrm{exp}\bigl[\bigl(x^{-\Theta} + y^{-\Theta}\bigr)^{-1/\Theta}\bigr]\mbox{,}$$ where $\Theta \in [0, \infty)$, $x = -\log u$, and $y = -\log v$. As $\Theta \rightarrow 0^{+}$, the copula limits to independence ($\mathbf{\Pi}$; P) and as $\Theta \rightarrow \infty$, the copula limits to perfect association ($\mathbf{M}$; M). The copula here is a bivariate extreme value copula ($BEV$), and parameter estimation for $\Theta$ requires numerical methods.

There are two other genetically related forms. Joe (2014, p. 197) describes an extension of the Galambos copula as a Galambos gamma power mixture (GLPM), which is Joe's BB4 copula, with the following form $$\mathbf{C}_{\Theta,\delta}(u,v) = \mathbf{GLPM}(u,v) = \biggl(x + y - 1 - \bigl[(x - 1)^{-\delta} + (y - 1)^{-\delta} \bigr]^{-1/\delta} \biggr)^{-1/\Theta}\mbox{,}$$ where $x = u^{-\Theta}$, $y = v^{-\Theta}$, and $\Theta \ge 0, \delta \ge 0$. (Joe shows $\delta > 0$, but zero itself seems to work without numerical problems in practical application.) As $\delta \rightarrow 0^{+}$, the “MTCJ family” (Mardia--Takahasi--Cook--Johnson) results (implemented internally with $\Theta$ as the incoming parameter). As $\Theta \rightarrow 0^{+}$ the Galambos above results with $\delta$ as the incoming parameter.

This second copula in turn has a lower extreme value limit form that leads to a min-stable bivariate exponential having Pickand dependence function of $$A(x,y; \Theta, \delta) = x + y - \bigl[x^{-\Theta} + y^{-\Theta} - (x^{\Theta\delta} + y^{\Theta\delta})^{-1/\delta} \bigr]^{-1/\Theta}\mbox{,}$$ where this third copula is $$\mathbf{C}^{LEV}_{\Theta,\delta}(u,v) = \mathbf{GLEV}(u,v) = \mathrm{exp}[-A(-\log u, -\log v; \Theta, \delta)]\mbox{,}$$ for $\Theta \ge 0, \delta \ge 0$ and is known as the two-parameter Galambos. (Joe shows $\delta > 0$, but $\delta = 0$ itself seems to work without numerical problems in practical application.)

Usage

GLcop(   u, v, para=NULL, ...)
GLEVcop( u, v, para=NULL, ...)
GLPMcop( u, v, para=NULL, ...) # inserts third parameter automatically
JOcopBB4(u, v, para=NULL, ...) # inserts third parameter automatically

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: To trigger $\mathbf{GL}(u,v)$, a vector (single element) of $\Theta$, to trigger $\mathbf{GLEV}(u,v)$, a two element vector of $\Theta$ and $\delta$ and alias is GLEVcop, and to trigger $\mathbf{GLPM}(u,v)$, a three element vector of $\Theta$, $\delta$, and any number (the presence of the third entry alone is the triggering mechanism) though aliases GLPM or JOcopBB4 will insert the third parameter automatically for convenience; and
...: Additional arguments to pass.

Author

W.H. Asquith

References