# Composition of a Single Symmetric Copula with Two Compositing Parameters (Khoudraji Device with Pi Independence) 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 [REMOVE_ME]$$ \mathbf{C}_{\alpha,\beta}(u,v) = u^\alpha v^\beta \cdot \mathbf{C}(u^{1-\alpha},v^{1-\beta})\mbox{.} [REMOVE_ME_2]$$ 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**). utf8 ## 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**). ```r 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. ## Returns 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. ## References Hofert, M., Kojadinovic, I., , 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., , J.F.: Springer, Cham, Switzerland, ISBN 978--3--319--64220--9, 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. ## Author(s) W.H. Asquith ## See Also `COP`, `breveCOP`, `composite2COP`, `composite3COP`, `convexCOP`, `glueCOP` ## Examples ```r ## 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) ```

composite1COP function