FGMcop: The Generalized Farlie--Gumbel--Morgenstern Copula

Description

The generalized Farlie--Gumbel--Morgenstern copula (Bekrizade et al., 2012) is $$\mathbf{C}_{\Theta, \alpha, n}(u,v) = \mathbf{FGM}(u,v) = uv[1 + \Theta(1-u^\alpha)(1-v^\alpha)]^n\mbox{,}$$ where $\Theta \in [-\mathrm{min}\{1, 1/(n\alpha^2)\}, +1/(n\alpha)]$, $\alpha > 0$, and $n \in 0,1,2,\cdots$. The copula $\Theta = 0$ or $\alpha = 0$ or $n = 0$ becomes the independence copula ($\mathbf{\Pi}(u,v)$; P). When $\alpha = n = 1$, then the well-known, single-parameter Farlie--Gumbel--Morgenstern copula results, and Spearman Rho (rhoCOP) is $\rho_\mathbf{C} = \Theta/3$ but in general $$\rho_\mathbf{C} = 12\sum_{r=1}^n {n \choose r} \Theta^r \biggl[\frac{\phantom{\alpha}\Gamma(r+1)\Gamma(2/\alpha)}{\alpha\Gamma(r+1+2/\alpha)} \biggr]^2 \mbox{.}$$ The support of $\rho_\mathbf{C}(\cdots;\Theta, 1, 1)$ is $[-1/3, +1/3]$ but extends via $\alpha$ and $n$ to $\approx [-0.50, +0.43]$, which shows that the generalization of the copula increases the range of dependency. The generalized version is implemented by FGMcop.

The iterated Farlie--Gumbel--Morgenstern copula (Chine and Benatia, 2017) for the $r$th iteration is $$\mathbf{C}_{\beta}(u,v) = \mathbf{FGMi}(u,v) = uv + \sum_{j=1}^{r} \beta_j\cdot(uv)^{[j/2]}\cdot(u'v')^{[(j+1)/2]}\mbox{,}$$ where $u' = 1-u$ and $v' = 1-v$ for $|\beta_j| \le 1$ that has $r$ dimensions $\beta = (\beta_1, \cdots, \beta_j, \cdots, \beta_r)$ and $[t]$ is the integer part of $t$. The copula $\beta = 0$ becomes the independence copula ($\mathbf{\Pi}(u,v)$; P). The support of $\rho_\mathbf{C}(\cdots;\beta)$ is approximately $[-0.43, +0.43]$. The iterated version is implemented by FGMicop. Internally, the $r$ is determined from the length of the $\beta$ in the para argument.

Usage

FGMcop( u, v, para=c(NA, 1,1), ...)
FGMicop(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 of parameters. For the generalized version, the $\Theta$, $\alpha$, and $n$ of the copula where the default argument shows the need to include the $\Theta$. However, if a fourth parameter is present, it is treated as a logical to reverse the copula ($u + v - 1 + \mathbf{FGM}(1-u,1-v; \Theta, \alpha, n)$). Also if a single parameter is given, then the $\alpha = n = 1$ are automatically set to produce the single-parameter Farlie--Gumbel--Morgenstern copula. For the iterated version, the $\beta$ vector of $r$ iterations;
...: Additional arguments to pass.

Author

W.H. Asquith

References

Bekrizade, Hakim, Parham, G.A., Zadkarmi, M.R., 2012, The new generalization of Farlie--Gumbel--Morgenstern copulas: Applied Mathematical Sciences, v. 6, no. 71, pp. 3527--3533.