# Archimedean Copulae Functions for ealuating densities of Archimedean copulae, generating random variates and fitting data to AC ```r dcopula.AC(u, theta, name = c("clayton", "gumbel"), log = TRUE) dcopula.clayton(u, theta, log = FALSE) dcopula.gumbel(u, theta, log = FALSE) rAC(name = c("clayton", "gumbel", "frank", "BB9", "GIG"), n, d, theta) rACp(name = c("clayton", "gumbel", "frank", "BB9", "GIG"), n, d, theta, A) rcopula.gumbel(n, theta, d) rcopula.clayton(n, theta, d) rcopula.frank(n, theta, d) rstable(n, alpha, beta = 1) rFrankMix(n, theta) rBB9Mix(n, theta) rcopula.Gumbel2Gp(n = 1000, gpsizes = c(2, 2), theta = c(2, 3, 5)) rcopula.GumbelNested(n, theta) fit.AC(Udata, name = c("clayton", "gumbel"), initial = 2, ...) ``` ## Arguments - `A`: `matrix`, dimension $d \times p$ containing asymmetry parameters. Rowsums must be equal to one. - `alpha`: `numeric`, parameter $0 < \alpha \le 2$, but c("$\\alpha\n$", "$ \\ne 1$"). - `beta`: `numeric`, parameter $-1 \le \beta \le 1$. - `d`: `integer`, dimension of copula. - `gpsizes`: `vector`, length of two, containing the group sizes. - `initial`: `numeric`, initial value used by `fit.AC()` in the call to `nlminb()`. - `log`: `logical`, whether log density values should be returned - `n`: `integer`, count of random variates. - `name`: `character`, name of copula. - `theta`: `vector`, copula parameter(s). - `u`: `matrix`, dimension $n \times d$, where d is the dimension of the copula and n is the number of vector values at which to evaluate density. - `Udata`: `matrix`, pseudo-uniform observations. - `...`: ellipsis, arguments are passed down to `nlminb()`. ## Details The function `dcopula.AC()` is a generic function, designed such that additional copulae, or expressions for densities of higher-dimensional copulae may be added. Clayton copula works in any dimension at present but Gumbel is only implemented for c("$d =\n$", "$ 2$"). To extend, one must calculate the d-th derivative of the generator inverse and take the logarithm of absolute value; this is the term called `loggfunc`. In addition, for other copulae, one needs the generator $phi$ and the log of the negative value of its first derivative `lnegphidash`. The random variates from `rAC()` with arbitrary dimension are generated by using the mixture construction of Marshall and Olkin. It may be used in place of the other functions `rcopula.clayton()`, `rcopula.gumbel()`, and `rcopula.frank()`. In addition, it allows simulation of BB9 and GIG copulas which don't have individual simulation routines. For the Clayton and Gumbel copulae, see page 192 and 222--224 in QRM. The random variates for the BB9 and Frank copula are obtained from a mixing distribution using a Laplace transform method (see page 224 of QRM). The function `rcopula.Gumbel2Gp()` generates sample from a Gumbel copula with two-group structure constructed using three Gumbel generators (see pages 222-224 and 227 of QRM). The function `rcopula.gumbelNested()` generates sample from a d-dimensional Gumbel copula with nested structure constructed using $(d-1)$ Gumbel generators. For the random variates of the Stable distribution, a default value $beta = 1$ is used; combined with a value for $alpha < 1$ yields a positive stable distribution, which is required for Gumbel copula generation; the case $alpha = 1$ has not been implemented. ## Returns vector or matrix in case of the density and random-generator related functions and a list object for the fitting function. ## See Also `nlminb` ## Examples ```r ## Gumbel r1 <- rAC("gumbel", n = 50, d = 7, theta = 3) head(r1) ## Weighted Gumbel alpha <- c(0.95,0.7) wtmatrix <- cbind(alpha, 1 - alpha) r2 <- rACp(name = "gumbel", n = 1000, d = 2, theta = c(4, 1), A = wtmatrix) head(r2) ## Gumbel with two-group structure r3 <- rcopula.Gumbel2Gp(n = 3000, gpsizes = c(3, 4), theta = c(2, 3, 5)) pairs(r3) ## Nested Gumbel r4 <- rcopula.GumbelNested(n=3000,theta=1:6) pairs(r4) ## Frank r5 <- rcopula.frank(1000, 2, 4) pairs(r5) ## Fitting of Gumbel and Clayton data(smi) data(ftse100) s1 <- window(ftse100, "1990-11-09", "2004-03-25") s1a <- alignDailySeries(s1) s2a <- alignDailySeries(smi) idx <- merge(s1a, s2a) r <-returns(idx) rp <- series(window(r, "1994-01-01", "2003-12-31")) rp <- rp[(rp[, 1] != 0) & (rp[, 2] !=0), ] Udata <- apply(rp, 2, edf, adjust = 1) mod.gumbel <- fit.AC(Udata, "gumbel") mod.clayton <- fit.AC(Udata, "clayton") mod.clayton ```

CopulaAC function