breveCOP: Add Asymmetry to a Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

Add Asymmetry to a Copula

Description

Adding permutation asymmetry (Chang and Joe, 2020, p. 1596) (isCOP.permsym) is simple for a bivariate copula family. Let \mathbf{C} be a copula with respective vectors of parameters \Theta_\mathbf{C}, then the permutation asymmetry is added through an asymmetry parameter \beta \in (-1, +1) by

\breve{\mathbf{C}}_{\beta;\Theta}(u,v) = v^{-\beta}\cdot\mathbf{C}(u, v^{(1+\beta)};\Theta)\mbox{, and}

for 0 \le \beta \le +1 by

\breve{\mathbf{C}}_{\beta;\Theta}(u,v) = u^{+\beta}\cdot\mathbf{C}(u^{(1-\beta)}, v;\Theta)\mbox{.}

The parameter \beta clashes in name and symbology with a parameter used by functions composite1COP, composite2COP, and composite3COP. As a result, support for alternative naming is provided for compatibility.

Usage

breveCOP(u,v, para, breve=NULL, ...)

Arguments

u	Nonexceedance probability u in the X direction;
v	Nonexceedance probability v in the Y direction;
para	A special parameter list (see Note);
breve	An alternative way from para to set the \beta for this function; and
...	Additional arguments to pass to the copula.

Value

Value(s) for the copula are returned.

Note

The following descriptions list in detail the structure and content of the para argument where cop1 and cop and para1 and para are respectively synonymous to have some structural similarity to the various copula constructors (compositors) of the copBasic package:

beta

— The \beta asymmetry parameter;

breve

— The \beta asymmetry parameter and presence of breve will not cause the non-use of beta; this feature is present so that beta remains accessible to the compositors that use beta (see Examples);

cop

— Function of the copula \mathbf{C};

cop1

— Alternative naming of the function of the coupla \mathbf{C};

para

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

para1

— Alternative naming of the vector of parameters \Theta_\mathbf{C} for \mathbf{C}.

The function silently restricts the \beta to its interval as defined, but parameter transform might be useful in some numerical optimization schemes. The following recipes might be useful for transform from a parameter in numerical optimization to the asymmetry parameter:

   #    transform into space for optimization
   BREVEtfunc <- function(p) { return(   qnorm((p[1] + 1) / 2) ) } # [-Inf, +Inf]
   # re-transform back into space for the copula
   BREVErfunc <- function(p) { return(2 * pnorm(p[1]) - 1)       } # [-1  , +1  ]

Author(s)

W.H. Asquith

References

Chang, B., and Joe, H., 2020, Copula diagnostics for asymmetries and conditional dependence: Journal of Applied Statistics, v. 47, no. 9, pp. 1587–1615, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/02664763.2019.1685080")}.

See Also

COP, convex2COP, convexCOP, composite1COP, composite2COP, composite3COP, FRECHETcop, glueCOP

Examples

para <- list(breve=0.24, cop1=FRECHETcop, para1=c(0.4, 0.56))
breveCOP(0.87, 0.35, para=para) # 0.282743

betas <- seq(-1,1, by=0.01)
bloms <- sapply(betas, function(b) {
             breveCOP(0.15, 0.25, para=list(cop=GLPMcop, para=c(2, 2), beta=b))
         } )
plot(betas, bloms, type="l", main="GLPMcop(u,v; 2,2) by breveCOP(beta)")

## Not run: 
  # Notice the argument cop and para name adjustments to show that
  # translation exists inside the function to have use flexibility.
  para <- list(beta=+0.44, cop1=FRECHETcop, para1=c(0.2, 0.56))
  UV   <- simCOP(1000, cop=breveCOP, para=para)
  para <- list(beta=-0.44, cop= FRECHETcop, para= c(0.2, 0.56))
  UV   <- simCOP(1000, cop=breveCOP, para=para) # 
## End(Not run)

## Not run: 
  # Testing on a comprehensive copula (Plackett)
  betas <- rhos <- thetas <- brhos <- NULL
  for(beta  in seq(-1, 1, by=0.1 )) {
    for(rho in seq(-1, 1, by=0.01))   {
       theta  <- PLACKETTpar(rho=rho, byrho=TRUE)
       thetas <- c(thetas, theta)
       para   <- list(  cop=PLcop,    para=theta, beta=beta)
       brho   <- rhoCOP(cop=breveCOP, para=para)
       betas  <- c(betas, beta); rhos <- c(rhos, rho)
       brhos  <- c(brhos, brho)
    }
  }
  df <- data.frame(beta=betas, theta=thetas, rho=rhos, brho=brhos)
  plot(df$theta, df$brho, log="x", pch=16, cex=0.9, col="seagreen",
       xlab="Plackett parameter", ylab="Spearman Rho")
  lines(df$theta[df$beta == 0], df$brho[df$beta == 0], col="red", lwd=2)
  # Red line is the Plackett in its permutation symmetric definition. #
## End(Not run)

## Not run: 
  # Here is an example for a test using mleCOP() to estimate a 5-parameter asymmetric
  # copula model to "some data" on transition from yesterday to today data for a very
  # large daily time series. The purpose of example here is to demonstrate interfacing
  # to the breveCOP() for it to add asymmetry to composition of two copula.
  myASYMCOP <- function(u,v, para, ...) {
    subpara <- list(alpha=para$alpha, beta=para$beta, cop1=GHcop, para1=para$para1,
                                                      cop2=PLcop, para2=para$para2)
    breveCOP(u,v, cop=convex2COP, para=subpara)
  }
  para <- list(alpha=+0.16934027, cop1=GHcop, para1=c(1.11144148, 10.32292673),
                beta=-0.01923808, cop2=PLcop, para2=3721.82966727)
  UV <- simCOP(30000, cop=myASYMCOP, para=para, pch=16, col=grey(0, 0.1))
  abline(0,1, lwd=3, col="red") #
## End(Not run)

## Not run: 
  # Here is a demonstration of the permutations of the passing of the
  # asymmetry parameter into the function and then by
  UV <- simCOP(1E3, cop=breveCOP, para=list(cop=HRcop, para=5), breve=+0.5)
  UV <- simCOP(1E3, cop=breveCOP, para=list(cop=HRcop, para=5), breve=-0.5)
  UV <- simCOP(1E3, cop=breveCOP, para=list(cop=HRcop, para=5,  beta =+0.5))
  UV <- simCOP(1E3, cop=breveCOP, para=list(cop=HRcop, para=5,  breve=+0.5))
  UV <- simCOP(1E3, cop=breveCOP, para=list(cop=HRcop, para=5,  beta=-0.4, breve=+0.5))

  para <- list(cop1=HRcop, para1=6, cop2=PSP, para2=NULL, alpha=1, beta=0.7)
  myCOP <- function(u,v, para, ...) breveCOP(u,v, cop=composite2COP, para=para)
  para$breve <- "here I am"
  UV <- simCOP(1E3, cop=composite2COP, para=para, seed=1) # breve is not used
  para$breve <- -0.16
  UV <- simCOP(1E3, cop=myCOP,         para=para, seed=1)
  para$breve <- +0.16
  UV <- simCOP(1E3, cop=myCOP,         para=para, seed=1) # 
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.