QRM: CopulaAC – R documentation

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CopulaAC

Archimedean Copulae

Description

Functions for ealuating densities of Archimedean copulae, generating random variates and fitting data to AC

Usage

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 < α ≤ 2, but α \ne 1.
`beta`	`numeric`, parameter -1 ≤ β ≤ 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 d = 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.

Value

vector or matrix in case of the density and random-generator related functions and a list object for the fitting function.

Examples

## 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

QRM

Provides R-Language Code to Examine Quantitative Risk Management Concepts

v0.4-31

GPL (>= 2)

Authors

Bernhard Pfaff [aut, cre], Marius Hofert [ctb], Alexander McNeil [aut] (S-Plus original (QRMlib)), Scott Ulmann [trl] (First R port as package QRMlib)

Initial release

2020-02-15