# Wrapper on a User-Level Formula to Become a Copula Function This function is intended to document and then to extend a simple API to end users to aid in implementation of other copulas for use within the `copBasic` package. There is no need or requirement to use `asCOP` for almost all users. However, for the mathematical definition of some copulas, the `asCOP` function might help considerably. This is because there is a need for special treatment of $u$ and $v$ vectors of probability as each interacts with the vectorization implicit in . The special treatment is needed because many copulas are based on the operators such as `min()` and `max()`. When numerical integration used by the `integrate()` function in in some copula operators, such as `tauCOP` for the **Kendall Tau** of a copula, special accommodation is needed related to the inherent vectorization in and how `integrate()` works. Basically, the problem is that one can not strictly rely in all circumstances on what does in terms of value recycling when $u$ and $v$ are of unequal lengths. The source code is straightforward. Simply put, if lengths of $u$ and $v$ are unity, then there is no concern, and even if the length of $u$ (say) is unity and $v$ is 21, then recycling of $u$ would often be okay. The real danger is when $u$ and $v$ have unequal lengths and those lengths are each greater than unity---the treatment can not be universally relied upon with the various numerics herein involving optimization and nested numerical integration. The example shows how a formula definition of a copula that is not a copula already implemented by `copBasic` is set into a function `deltacop` and then used inside another function `UsersCop` that will be the official copula that is compatible with a host of functions in `copBasic`. The use of `asCOP` provides the length check necessary on $u$ and $v$, and the argument `...` provides optional parameter support should the user's formula require more settings. utf8 ```r asCOP(u, v, f=NULL, ...) ``` ## Arguments - `u`: Nonexceedance probability $u$ in the $X$ direction; - `v`: Nonexceedance probability $v$ in the $Y$ direction; - `f`: A function for which the user desires to make as a copula; and - `...`: Additional arguments to pass to the function `f` (such as parameters, if needed, for the copula in the form of a list). ## Returns The value(s) for the copula are returned. ## References Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p. ## Author(s) W.H. Asquith ## See Also `COP` ## Examples ```r ## Not run: # Concerning Nelsen (2006, exer. 3.7, pp. 64--65) "trianglecop" <- function(u,v, para=NULL, ...) { # If para is set, then the triangle is rotated 90d clockwise. if(! is.null(para) && para == 1) { t <- u; u <- v; v <- t } if(length(u) > 1 | length(v) > 1) stop("only scalars for this function") v2<-v/2; if(0 <= u & u <= v2 & v2 <= 1/2) { return(u ) } else if(0 <= v2 & v2 < u & u < 1-v2) { return(v2 ) } else if(1/2 <= 1-v2 & 1-v2 <= u & u <= 1 ) { return(u+v-1) } else { stop("should not be here in logic") } } "UsersCop" <- function(u,v, ...) { asCOP(u,v, f=trianglecop, ...) } n=20000; UV <- simCOP(n=n, cop=UsersCop) # The a-d elements of the problem now follow: # (a) Pr[V = 1 - |2*U -1|] = 1 and Cov(U,V) = 0; so that two random variables # can be uncorrelated but each is perfectly predictable from the other mean(UV$V - (1 - abs(2*UV$U -1))) # near zero; Nelsen says == 0 cov(UV$U, UV$V) # near zero; Nelsen says == 0 # (b) Cop(m,n) = Cop(n,m); so that two random variables can be identically # distributed, uncorrelated, and not exchangeable EMPIRcop(0.95,0.17, para=UV) # = A EMPIRcop(0.17,0.95, para=UV) # = B; then A != B # (c) Pr[V - U > 0] = 2/3; so that two random variables can be identically # distributed, but their difference need not be symmetric about zero tmp <- (UV$V - UV$U) > 0 length(tmp[tmp == TRUE])/n # about 2/3; Nelsen says == 2/3 # the prior two lines yield about 1/2 for independence copula P() # (d) Pr[X + Y > 0] = 2/3; so that uniform random variables on (-1,1) can each # be symmetric about zero, but their sum need not be. tmp <- ((2*UV$V - 1) + (2*UV$U - 1)) > 0 length(tmp[tmp == TRUE])/n # about 2/3; Nelsen says == 2/3 ## End(Not run) ## Not run: # Concerning Nelsen (2006, exam. 3.10, p. 73) "shufflecop" <- # assume scalar arguments for u and v function(u,v, para, ...) { m <- para$mixer; subcop <- para$subcop if(is.na(m) | m <= 0 | m >= 1) stop("m ! in [0,1]") if(u <= m) { return( subcop(1-m+u, v, para=para$para) - subcop(1-m, v, para=para$para)) } else { return(v - subcop(1-m, v, para=para$para) + subcop(u-m, v, para=para$para)) } } "UsersCop" <- function(u,v, para=NULL) { asCOP(u,v, f=shufflecop, para=para) } n <- 1000; u <- runif(n) para <- list(mixer=runif(1), subcop=W, para=20) v <- sapply(1:n, function(i) { simCOPmicro(u[i], cop=UsersCop, para=para) } ) plot(data.frame(U=u, V=v), pch=17, col=rgb(1,0,1,1), xlab="U, NONEXCEEDANCE PROBABILTY", ylab="V, NONEXCEEDANCE PROBABILITY") mtext("Shuffle Copula Nelsen (2006, exam. 3.10, p. 73)") # Concerning Nelsen (2006, exam. 5.14, p. 195) "deltacop" <- function(u,v, ...) { min(c(u,v,(u^2+v^2)/2)) } "UsersCop" <- function(u,v, ...) { asCOP(u,v, f=deltacop, ...) } isCOP.PQD(cop=UsersCop) # TRUE + Rho=0.288 and Tau=0.333 as Nelsen says isCOP.LTD(cop=UsersCop, wrtV=TRUE) # FALSE as Nelsen says isCOP.RTI(cop=UsersCop, wrtV=TRUE) # FALSE as Nelsen says ## End(Not run) ```

asCOP function