# The Generalized Farlie--Gumbel--Morgenstern Copula The **generalized Farlie--Gumbel--Morgenstern copula** (Bekrizade **et al.**, 2012) is [REMOVE_ME]$$ \mathbf{C}_{\Theta, \alpha, n}(u,v) = \mathbf{FGM}(u,v) = uv[1 + \Theta(1-u^\alpha)(1-v^\alpha)]^n\mbox{,} [REMOVE_ME_2]$$ 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 [REMOVE_ME]$$ \rho_\mathbf{C} = 12\sum_{r=1}^n {n \choose r} \Theta^r \biggl[\frac{\phantom{\alpha}\Gamma(r+1)\Gamma(2/\alpha)}{\alpha\Gamma(r+1+2/\alpha)} \biggr]^2\mbox{.} [REMOVE_ME_2]$$ 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 [REMOVE_ME]$$ \mathbf{C}_{\beta}(u,v) = \mathbf{FGMi}(u,v) = uv + \sum_{j=1}^{r} \beta_j\cdot(uv)^{[j/2]}\cdot(u'v')^{[(j+1)/2]}\mbox{,} [REMOVE_ME_2]$$ 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. utf8 ## 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. ```r FGMcop( u, v, para=c(NA, 1,1), ...) FGMicop(u, v, para=NULL, ...) ``` ## 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. ## Returns Value(s) for the copula are returned. ## 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. Chine, Amel, and Benatia, Fatah, 2017, Bivariate copulas parameters estimation using the trimmed L-moments methods: Afrika Statistika, v. 12, no. 1, pp. 1185--1197. ## Author(s) W.H. Asquith ## See Also `P`, `mleCOP` ## Examples ```r ## Not run: # Bekrizade et al. (2012, table 1) report for a=2 and n=3 that range in # theta = [-0.1667, 0.1667] and range in rho = [-0.1806641, 0.4036458]. However, # we see that they have seemingly made an error in listing the lower bounds of theta: rhoCOP(FGMcop, para=c( 1/6, 2, 3)) # 0.4036458 rhoCOP(FGMcop, para=c( -1/6, 2, 3)) # Following error results # In cop(u, v, para = para, ...) : parameter Theta < -0.0833333333333333 rhoCOP(FGMcop, para=c(-1/12, 2, 3)) # -0.1806641 ## End(Not run) ## Not run: # Support of FGMrcop(): first for r=1 iterations and then for large r. sapply(c(-1, 1), function(t) rhoCOP(cop=FGMicop, para=rep(t, 1)) ) # [1] -0.3333333 0.3333333 sapply(c(-1, 1), function(t) rhoCOP(cop=FGMicop, para=rep(t,50)) ) # [1] -0.4341385 0.4341385 ## End(Not run) ## Not run: # Maximum likelihood estimation near theta upper bounds for a=3 and n=2. set.seed(832) UV <- simCOP(3000, cop=FGMcop, para=c(+0.16, 3, 2)) # Define a transform function for parameter domain, though mleCOP does # provide some robustness anyway---not forcing n into the positive # domain via as.integer(exp(p[3])) seems to not always be needed. FGMpfunc <- function(p) { d <- p[1]; a <- exp(p[2]); n <- as.integer(exp(p[3])) lwr <- -min(c(1,1/(n*a^2))); upr <- 1/(n*a) d <- ifelse(d <= lwr, lwr, ifelse(d >= upr, upr, d)) return( c(d, a, n) ) } para <- c(0.16, 3, 2); init <- c(0, 1, 1) ML <- mleCOP(UV$U, UV$V, cop=FGMcop, init.para=init, parafn=FGMpfunc) print(ML$para) # [1] 0.1596361 3.1321228 2.0000000 # So, we have recovered reasonable estimates of the three parameters # given through MLE estimation. densityCOPplot(cop=FGMcop, para= para, contour.col="red") densityCOPplot(cop=FGMcop, para=ML$para, ploton=FALSE) # ## End(Not run) ```

FGMcop function