copBasic (version 1.5.1)

PlackettPlackettABKGtest: Parameters and L-comoments of a Composition of Two Plackett Copulas

Description

These data contain the stochastically generated parameter space of a Plackett-Plackett composited copulas to support a fast lookup of the relation between four copula parameters and the four L-comoments of L-correlation and L-coskew---hence, support the copulatic method of L-comoments. This particular data set is provided in the copBasic package for self-contained parameter estimation, but much much greater data sets are available from the author.

Specifically, these data contain the stochastically generated parameter space of the $\alpha$ and $\beta$ mixing parameters of the composited copula in which the two copulas to composite $\mathbf{A}$ and $\mathbf{B}$ are set as Plackett types.

The construction of an asymmetric copula from composition two copulas provide for more sophisticated structures of dependence between variables. Let $\mathbf{A}$ and $\mathbf{B}$ be copulas with their own parameter sets. Then

$$\mathbf{C}_{\alpha,\beta,\kappa,\gamma}(u,v) = u^\kappa v^\gamma \mathbf{A}([u^{1-\kappa}]^\alpha, [v^{1-\gamma}]^\beta) \cdot \mathbf{B}([u^{1-\kappa}]^{1-\alpha},[v^{1-\gamma}]^{1-\beta})\mbox{,}$$ defines a family of copulas $\mathbf{C}_{\alpha,\beta,\kappa,\gamma}$, with parameters $\alpha,\beta,\kappa,\gamma \in \mathcal{I}:[0,1]$.

The first Plackett copula $\mathbf{A}$ (PLACKETTcop) was randomly generated between the Frechet-Hoeffding lower bound copula $\mathbf{W}$ (W) and the Frechet-Hoeffding upper bound copula $\mathbf{M}$ (M). The second Plackett copula $\mathbf{B}$ (PLACKETTcop) was randomly generated in a similar fashion as $\mathbf{A}$.

To further clarify, $\mathbf{A}_\Theta$ parameter is on the interval $[0,\infty]$ and the $\mathbf{B}_\Theta$ parameter is on the inteval $[0,\infty]$. In reality the $\Theta$ parameters were generated uniformly in log-space and then transformed. The base-10 log of $\mathbf{A}_\Theta$ was on the interval $[-5,5]$, and the log of $\mathbf{B}_\Theta$ was on the interval $[-5,5]$. (See such limits of generation in the Source section of this documentation.)

Usage

data(PlackettPlackettABKGtest)

source

This data set was created using the simcompositeCOP function, which in turn defaults to use of the composite2COP function for compositing of the two Plackett copulas. The usage is shown below. mainpara <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, para1gen=function() { return(10^runif(1, min=-5, max=5)) }, para2gen=function() { return(10^runif(1, min=-5, max=5)) })

nsim <- 20500 sample.size.for.estimation <- 1000 PlackettPlackettABKGtest <- simcomposite3COP(n=sample.size.for.estimation, nsim=nsim, parent=mainpara) save(PlackettPlackettABKGtest, file="PlackettPlackettABKG.RData", compress="xz") For the generation process, the mainpara list specifies the two copulas to use in the composite, and two generation functions for the parameters of the respective copulas are housed in para1gen and para2gen. For each of 20500 simulations of size 1000 of the stochastically generated Plackett-Plackett composited copulas, the L-correlation and L-coskew matrices are computed. The opposing diagonals of the L-correlation and L-coskew of the matrices are requested internally by simcompositeCOP and shown. The L-comoment matrices are computed by the lcomoms2() function of the lmomco package.

References

Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.

Serfling, R., and Xiao, P., 2007, A contribution to multivariate L-moments---L-comoment matrices: Journal of Multivariate Analysis, v. 98, pp. 1765--1781.

Examples

data(PlackettPlackettABKGtest)
summary(PlackettPlackettABKGtest)

PlackettPlackettABKG <- as.data.frame(PlackettPlackettABKGtest)
# Let us now visualize the parameter space
plot.complcom.space <- function(text=NULL) {
  U12 <- PlackettPlackettABKG$T2.12
  U21 <- PlackettPlackettABKG$T2.21
  V12 <- PlackettPlackettABKG$T3.12
  V21 <- PlackettPlackettABKG$T3.21
  U <- c(U12,U21); V <- c(V12,V21)
  plot(U,V, type="n", xlim=c(-1,1), ylim=c(-0.2,0.4),
       xlab="L-CORRELATION",
       ylab="L-COSKEW")
  # plot transparent blue L-correlation and L-coskew first
  points(U12,V12, col=rgb(   0, 0, 1, 0.12), pch=16)
  # plot transparent purple L-correlation and L-coskew second
  points(U21,V21, col=rgb(0.85, 0, 1, 0.12), pch=16)
  abline(v=0); abline(h=0) # cross lines
  mtext(text)
}
my.text <- "L-COMOMENTS OF ABGK COMPOSITED PLACKETT-PLACKETT COPULA"
plot.complcom.space(text=my.text) # plot the parameter space


# Let us now visualize the parameter space
plot.complcom.space2 <- function(text=NULL) {
  U12 <- PlackettPlackettABKG$T3.12
  U21 <- PlackettPlackettABKG$T3.21
  V12 <- PlackettPlackettABKG$T4.12
  V21 <- PlackettPlackettABKG$T4.21
  U <- c(U12,U21); V <- c(V12,V21)
  plot(U,V, type="n", xlim=c(-0.2,0.4), ylim=c(-0.2,0.2),
       xlab="L-COSKEW",
       ylab="L-COKURTOSIS")
  # plot transparent blue L-correlation and L-coskew first
  points(U12,V12, col=rgb(   0, 0, 1, 0.12), pch=16)
  # plot transparent purple L-correlation and L-coskew second
  points(U21,V21, col=rgb(0.85, 0, 1, 0.12), pch=16)
  abline(v=0); abline(h=0) # cross lines
  mtext(text)
}
my.text <- "L-COMOMENTS OF ABKG COMPOSITED PLACKETT-PLACKETT COPULA"
plot.complcom.space2(text=my.text) # plot the parameter space