composite1COP: Composition of a Single Symmetric Copula with Two Compositing...

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

Composition of a Single Symmetric Copula with Two Compositing Parameters (Khoudraji Device with Pi Independence)

Description

The composition of a single copula (Salvadori et al., 2006, p. 266, prop. C.3) is created by the following result related to “composition of copulas” in that reference. This construction technique is named the Khoudraji device within the copula package (see khoudrajiCopula therein) (Hofert et al., 2018, pp. 120–121). Suppose \mathbf{C}(u,v) is a symmetric copula (see COP) with parameters \Theta and \mathbf{C} \ne \mathbf{\Pi} (for \mathbf{\Pi} see P), then a family of generally asymmetric copulas \mathbf{C}_{\alpha,\beta; \Theta} with two compositing parameters 0 < \alpha,\beta < 1, and \alpha \ne \beta, which also includes just the copula \mathbf{C}(u,v) as a limiting case for \alpha = \beta = 0 and is given by

\mathbf{C}_{\alpha,\beta}(u,v) = u^\alpha v^\beta \cdot \mathbf{C}(u^{1-\alpha},v^{1-\beta})\mbox{.}

The composite1COP function provides the means for inserting permutation asymmetry from a permutation symmetric copula as described by Joe (2017, p. 124), but do so in a more general way through the provision of two and not just one parameter. Joe's description is supported herein if one of the \alpha or \beta is held at zero. Very loosely, the \alpha > 0 kicks probability density down towards the lower right corner, whereas \beta > 0 kicks density up towards the upper left corner. Finally, the composite2COP function is based on a slighty more general result (see composite2COP for further details of copula composition and more contextualization of Hofert et al. (2018) remarks on the Khoudraji device).

Usage

composite1COP(u, v, para, ...)
khoudraji1COP(u, v, para, ...)
khoudrajiPCOP(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 composited copula are returned.

Note

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

alpha

— The \alpha compositing parameter;

beta

— The \beta compositing parameter;

cop1

— Function of the copula \mathbf{C}(u,v); and

para1

— Vector of parameters \Theta_\mathbf{C} for \mathbf{C}(u,v).

For the para argument, the same nomenclature as used for composite2COP is used with obviously cop2 and para2 dropped for composite1COP. The cop1 and para1 names remain enumerated for composite1COP so that the para argument of the more general composite2COP function could be used directly in composite1COP. Albeit, the second copula and its parameters would not be used. A more complex (extended) composition in composite3COP extends this basic parameter structure.

Author(s)

W.H. Asquith

References

Hofert, M., Kojadinovic, I., Mächler, M., and Yan, J., 2018, Elements of copula modeling with R: Dordrecht, Netherlands, Springer.

Joe, H., 2017, Parametric copula families for statistical models (chap. 8) in Copulas and dependence models with applications—Contributions in honor of Roger B. Nelsen, eds. Flores, U.M., Amo Artero, E., Durante, F., Sánchez, J.F.: Springer, Cham, Switzerland, ISBN 978–3–319–64220–9, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-3-319-64221-5")}.

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

See Also

COP, breveCOP, composite2COP, composite3COP, convexCOP, glueCOP

Examples

## Not run: 
alpha <- 0.24; beta <- 0.23; Theta1 <- NA;
# W() does not use a parameter, but show how a parameter would be set if needed.
para  <- list(alpha=alpha, beta=beta, cop1=W, para1=Theta1)
t <- composite1COP(0.4, 0.6, para)
if( t != W(0.4, 0.6)) message("Not equal as expected") #
## End(Not run)

## Not run: 
  # Hofert et al. (2018, p. 124, eq. 3.15)
  #   No matter what copula is chosen, Kendall tau must be
  #     Tau <= (alpha*beta)/(alpha + beta - alpha*beta)
  #   and those authors report Tau <= 0.5816. We can test this computation by
  para <- list(cop=M, para=NULL, alpha=1-0.6, beta=1-0.95)
  tauCOP(khoudrajiPCOP, para=para) # 0.5816283 
## End(Not run)

## Not run: 
  # Next use this as a chance to check logic flow through the various
  # "compositing" operators and their as needed dispatch to COP().
  my.para <- list(cop1=GHcop, para1=exp(+1.098612) + 1,
                  cop2=PLcop, para2=exp(-1.203973),
                  alpha=0.5,  beta=0.25, kappa=0.1, gamma=0.1,
                  weights=c(0.95, 0.05))
  # uses cop1/2, para1/2, only weights
  nustarCOP(cop=convexCOP,     para=my.para) # 0.8570434

  # uses cop1/2, para1/2, only alpha
  nustarCOP(cop=convex2COP,    para=my.para) # 0.2697063

  # uses cop1,   para1,   only alpha / beta
  nustarCOP(cop=composite1COP, para=my.para) # 0.5103119

  # uses cop1/2, para1/2, only alpha / beta
  nustarCOP(cop=composite2COP, para=my.para) # 0.0714571

  # uses cop1/2, para1/2, only alpha, beta, kappa, gamma
  nustarCOP(cop=composite3COP, para=my.para) # 0.0792634 
## End(Not run)

## Not run: 
  # Hofert et al. (2018, p. 121, fig. 3.20, left panel)
  #   The ordering of copula and the "1-" operations on alpha and beta in copBasic
  #   differ from that shown in Hofert et al. (2018), but instead of their
  #   "kho(0.6, 0.95)(CLcop(6), P)" notation for the their left panel, in copBasic
  #   we can reproduce their simulation by the following. So, swapping notation
  #   between the copula package (khoudarjiCopula) and copBasic would be required.
  para <- list(cop=CLcop, para=6, alpha=1-0.95, beta=1-0.6)
  UV <- simCOP(n=5000, cop=khoudrajiPCOP, para=para) # 
## End(Not run)

wasquith/copBasic documentation built on Feb. 17, 2025, 11:39 a.m.