convex2COP: Convex Combination of Two Copulas

In copBasic: General Bivariate Copula Theory and Many Utility Functions

Convex Combination of Two Copulas

Description

The convex composition of two copulas (Joe, 2014, p. 155) provides for some simple complexity extension between copula families. Let \mathbf{A} and \mathbf{B} be copulas with respective vectors of parameters \Theta_\mathbf{A} and \Theta_\mathbf{B}, then the convex combination of these copulas is

\mathbf{C}^{\times}_{\alpha}(u,v) = \alpha\cdot\mathbf{A}(u, v; \Theta_\mathbf{A}) - (1-\alpha)\cdot\mathbf{B}(u,v; \Theta_\mathbf{B})\mbox{,}

where 0 \le \alpha \le 1. The generalization of this function for N number of copulas is provided by convexCOP.

Usage

convex2COP(u,v, para, ...)

Arguments

u	Nonexceedance probability u in the X direction;
v	Nonexceedance probability v in the Y direction;
para	A special parameter list (see Note); and
...	Additional arguments to pass to the copula.

Value

Value(s) for the convex combination copula is returned.

Note

The following descriptions list in detail the structure and content of the para argument:

alpha

— The \alpha compositing parameter;

cop1

— Function of the first copula \mathbf{A};

cop2

— Function of the second copula \mathbf{B};

para1

— Vector of parameters \Theta_\mathbf{A} for \mathbf{A}; and

para2

— Vector of parameters \Theta_\mathbf{B} for \mathbf{B}.

Author(s)

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

See Also

COP, breveCOP, convexCOP, composite1COP, composite2COP, composite3COP, FRECHETcop,
glueCOP

Examples

para <- list(alpha=0.24, cop1=FRECHETcop, para1=c(0.4, 0.56),
                         cop2=PSP,        para2=NA)
convex2COP(0.87,0.35, para=para) # 0.3188711
## Not run: 
# Suppose we have a target Kendall Tau of 1/3 and a Gumbel-Hougaard copula seems
# attractive but the GH has just too much upper tail dependency for comfort. We
# think from data analysis that an upper tail dependency that is weaker and near
# 2/10 is the better. Let us convex mix in a Plackett copula and optimize.
TargetTau <- tauCOP(cop=GHcop, para=1.5) # 1/3 (Kendall Tau)
taildepCOP(   cop=GHcop, para=1.5, plot = TRUE)$lambdaU  # 0.4126
TargetUpperTailDep <- 2/10

# **Serious CPU time pending for this example**
par <- c(-.10, 4.65) # Initial guess but the first parameter is in standard
# normal for optim() to keep us in the [0,1] domain when converted to probability.
# The guesses of -0.10 (standard deviation) for the convex parameter and 4.65 for
# the Plackett are based on a much longer search times as setup for this problem.
# The simplex for optim() is going to be close to the solution on startup.
"afunc" <- function(par) {
   para <- list(alpha=pnorm(par[1]), cop1=GHcop,       para1=1.5,
                                     cop2=PLACKETTcop, para2=par[2])
   tau  <- tauCOP(cop=convex2COP, para=para)
   taildep <- taildepCOP(cop=convex2COP, para=para, plot = FALSE)$lambdaU
   err <- sqrt((TargetTau - tau)^2 + (TargetUpperTailDep - taildep)^2)
   print(c(pnorm(par[1]), par[2], tau, taildep, err))
   return(err)
}
mysolution <- optim(par, afunc, control=list(abstol=1E-4))

para <- list(alpha=.4846902, cop1=GHcop,       para1=1.5,
                             cop2=PLACKETTcop, para2=4.711464)
UV <- simCOP(n=2500, cop=convex2COP, para=para, snv=TRUE) #
## End(Not run)

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