FGMcop: The Generalized Farlie-Gumbel-Morgenstern Copula

In copBasic: General Bivariate Copula Theory and Many Utility Functions

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 rth 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,       ...)

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.

Value

Value(s) for the copula are returned.

Author(s)

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.

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.

See Also

P, mleCOP

Examples

## 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)

copBasic documentation built on June 28, 2025, 1:08 a.m.